Is there any place in the Bible that Jesus Himself claims that all believers are members of His Body? Is there any place in the Bible where Jesus Himself claims that all believers are members of His Body? Or is it something that we only know from His apostles?

I mean we can infer this thought from such words of Jesus like "I am the vine and you are the branches" or "if the grain of wheat dies, it will bring a of of fruit, however, Jesus also spoke about His body at the last supper, so He might have said something about His believers being members of His Body even prior to His crucifixion.
Unless you do not consider Paul's testimony of his conversion as Jesus's direct claim, then no, as the disciples are not ready yet to receive such revelation.

When Saul was persecuting the church, on his way to Damascus, the Lord Jesus asked him, "Saul, Saul, why are you persecuting Me?" (Acts 9:4). The Lord’s word was quite strange. He did not say, "Saul, Saul, why are you persecuting My disciples," but "Saul, Saul, why are you persecuting Me?" He did not say, "Saul, Saul, why are you persecuting My church,” but “Saul, Saul, why are you persecuting Me?" 

This showed and impressed Paul that the church and Christ are one. Christ is saying "When you [Paul] persecuted My followers, you persecuted My Body. To persecute My Body is to persecute Me." Therefore, the "Me" in 9:4 is corporate, comprising the Lord Jesus and all His believers.

Saul's experience of the corporate Me, Christ and the Body, must have made a deep impression on him and affected his future ministry regarding Christ and the church. That experience laid the foundation for his ministry.

EDIT: WE can and should infer from John 15 and 12:24 that there is a perichoretic union between Christ and the believers. Which is similar to the reality of Christ's relationship with His member of His Body.

George E. Ladd comments:


  The idiom of abiding is usually called mysticism, but it is difficult
  to define. There is a mutual abiding of the believer in Christ (16:56;
  14:20, 21; 15:5; 17:21) and Christ in the believer (6:56; 14:20, 23;
  15:5; 17:23, 26). This is analogous to the Son abiding in the Father
  (10:38, 14:10, 11, 20, 21; 17:21) and the Father abiding in the Son
  (10:38, 14:10, 11, 21; 17:21, 23). Once it is said that believers are in both
  the Father and the Son (17:21); and once it is said that both Father
  and Son will come to make their abode in believers (Theology of the New Testament)


G.B. Verify boldly says the meaning of "abide" is "coninherence", the synonyms for "perichoresis" (Life in Christ).