A common argument made by atheists against Jesus and the Bible is that not only is Jesus not God, but He never said He is. Therefore, they say this is a false claim made by Christians that Jesus Himself never made. First of all, why would we look to those who deny the existence of God, the deity of Jesus, and the validity of the Bible for the right interpretation of scripture? How are those who appoint themselves as the arbiters of truth and yet deny absolute truth in a better position to tell everyone else what to believe? The source of their alleged authority is themselves. Man does not have a great track record of being his own authority, and this goes all the way back to the garden of Eden (Gen 3:1-7). Things are either based on the Bible, or they are made up by man. The enemies of Jesus challenged His authority continually. They challenged by what authority He cast out demons (Mt 9:33-34). He later responded to their challenge partly by asking them by what authority their children did so (Mt 12:22-37). Of course, they had no answer. Shortly before they arrested Him, falsely accused Him, and sent Him to His death, they again questioned His authority (Mt 21:23-27). He answered their question with a question, partly because they were not sincerely asking, and also to turn the tables back on them. He told them He would answer their question if they answered His. He asked them by whose authority John baptized. Was it by God or man? They refused to answer because they realized no matter which way they answered, they would be exposed. Jesus said He would neither answer their question.
These questions get to the heart of the matter. Atheists are more accurately deniers because in their hearts they know there is a God (Rom 1:18-32). The question they pose is supposedly about the existence of God, but that is not really the issue. The question is, are we submitting to the authority God has over us? He is Creator and Lord, and He purchased us from sin with His blood (Gen 1, Ps 100:23, Acts 20:28). I suggest articles and videos on this website as well as the book "Why Believe?" for further discussion on the validity of the Bible as the word of God.
Now to address the specific question about whether or not Jesus said He is God. The Bible clearly asserts throughout that Jesus is God. For further study on that, you can review and article and video on this website entitled "The Deity of Jesus". Since the Bible says Jesus is God and the Bible is the word of God, anything the Bible says about Jesus is Jesus telling us about Himself. However, let us look specifically at things Jesus is recorded as having said while He was here on earth to see if He ever asserted that He is God.
The book of John has many themes, but probably the overarching theme throughout it is the deity of Jesus. Jesus asserted clearly and repeatedly that He is God. He told Nathaniel, "Verily, verily, I say unto you, Hereafter ye shall see heaven open, and the angels of God ascending and descending upon the Son of man (Jn 1:51)". Jesus is in authority over the angels and is superior to them (Mt 13:41, Mt 24:31, Mt 26:35, Heb 1). This includes the fallen angels we call demons. Even the demons and their servants the human enemies of Jesus recognized this (Mt 9:34, Mk 1:34, Mk 5:12, Lk 4:41). The angels were created by and are subject to God alone (Ps 103:20). This means that Jesus said and showed He is God by asserting His authority over them.
Jesus said he He would raise Himself from the dead (Jn 2:19-21, Jn 10:17-18). This is clearly an assertion of deity. Jesus told Nicodemus that He was on earth and in heaven at the same time (Jn 3:13). He told the woman at the well that He is the One who gives eternal life (Jn 4:14). He also told her that He is the Messiah (Jn 4:25-26). He also told the priests this (Mk 14:61-62). The prophecies in the Old Testament about the Messiah clearly show that He is God (Ps 110, Is 9:6, Eze 13 and 14, Dan 7, Mic 5:2, etc.). The enemies of Jesus understood that He was asserting His deity (Jn 5:17-18, Jn 10:30-33 & 37-38). He demonstrated His authority over nature by healing the sick, raising the dead, walking on water, calming storms, turning water into wine, and multiplying food. He was not the only One to perform miracles, but there are at least two ways His miracles are unique. Nobody even comes close the volume of miracles Jesus did (Jn 7:31, Jn 15:24, Jn 20:30-31, Jn 21:25), and He did miracles by His own authority because He is God in the flesh. The Jews demanded that He tell them plainly whether or not He is the Christ (Jn 10:24-26). He replied that His miracles testify to that. He said that the problem was not His failure to demonstrate who He is, but their failure to believe the evidence He so plainly set before them (Jn 10:37-38). Jesus said He has the authority to raise the dead (Jn 5:22-30, Jn 6:44-48). He said He came down from heaven (Jn 6:38). He said He is the One who gives the Holy Ghost (Jn 7:37-39, Jn 14:18, Jn 15:26, Jn 16:7). He said He is the Father manifested in the flesh (Jn 10:30, Jn 14:7-11). He said, "I and my Father are one (Jn 10:30)". Notice the grammar of this statement. He did not say, "My Father and I". He said, "I and my Father". The Jews understood He was asserting His deity and attempted to kill Him for it. Also, the word my is in italics is the KJV because it was added by the translators. Jesus said, "I and Father are one". The word one in this verse is the Greek word heis, which means the numeral one. It does not mean unity. Jesus was not saying He was on the same page as the Father. He was saying He IS the Father.
We should not be confused by statements of Jesus which mention both He and Father. He is not talking about the Father as a separate divine person, nor is He saying that He is not divine. He is saying that He is God manifested in the flesh. The distinction is between divinity and humanity. Jesus is both in One. Sometimes He spoke from the reference point of His deity, and sometimes He spoke from the reference point of His humanity. This also shows that He is uniquely God in the flesh. Though the Lord fills His believers with His Spirit, He does not make them deity. The relationship between the deity of Jesus and the humanity He was manifested in is unique to Him alone (Jn 3:13 & 34). Not even the angels can claim this unique identity (Heb 1).
Believers are granted a special status with Him, but are not equal to Him. None is.
Jesus said He is God in every way. He said it by fulfilling His own word in the Old Testament concerning His birth, life, death, and resurrection. He said it by the perfect life He lived before all (Jn 8:40, Jn 10:32). He said it by the miracles which He did. He said it by rising from the dead on His own. He said it by the lives He changed. He said it by forgiving sins (Mk 2:1-12). He said it by promising and giving the Holy Ghost. He said it by His teaching. He said it by the claims He made and backed up. He said it directly and repeatedly. He gave us the written word of the New Testament. Throughout the last 2,000 years, He continues to demonstrate who He is. He gave us more than enough proof to justify believing in Him. No man will have an excuse for their refusal to believe in Him (Rom 1:18-32). He will help us with our struggles, but He will not excuse those who steadfastly refuse to believe (Acts 17:22-31). We will all give account to Him at the end. Those who rejected Him will have to face how He gave them every chance to be saved. Perhaps that will be the greatest torment of eternity without Him. He died so we could avoid that final outcome and have eternal life with Him (Jn 3:16). He told the apostles what to preach (Mt 28:18-20, Mk 16:15-20, Lk 24:46-49, Jn 20:21-23), and they preached it (Acts 2:38-29, Acts 8:12-17, Acts 10:43-48, Act 19:1-7). They also recorded the teaching of Jesus and theirs in the New Testament for us. All He requires is faith which leads to obedience of His word.