Ephesians 4:3-6: "3 eager to maintain the unity of the Spirit in the bond of peace. 4 There is one body and one Spirit - just as you were called to the one hope that belongs to your call - 5 one Lord, one faith, one baptism, 6 one God and Father of all, who is over all and through all and in all."
Romans 12:4-5: "4 For as in one body we have many members, and the members do not all have the same function, 5 so we, though many, are one body in Christ, and individually members one of another."
John 17:11: "'And I am no longer in the world, but they are in the world, and I am coming to you. Holy Father, keep them in your name, which you have given me, that they may be one, even as we are one.'"
John 17:20-23: "20 'I do not ask for these only, but also for those who will believe in me through their word, 21 that they may all be one, just as you, Father, are in me, and I in you, that they also may be in us, so that the world may believe that you have sent me. 22 The glory that you have given me I have given to them, that they may be one even as we are one, 23 I in them and you in me, that they may become perfectly one, so that the world may know that you sent me and loved them even as you loved me.'"
Psalm 133: "1 Behold, how good and pleasant it is when brothers dwell in unity! 2 It is like the precious oil on the head, running down on the beard, on the beard of Aaron, running down on the collar of his robes! 3 It is like the dew of Hermon, which falls on the mountains of Zion! For there the LORD has commanded the blessing, life forevermore."
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The text is public domain:
Blest be the tie that bindsOur hearts in Christian love;The fellowship of kindred mindsIs like to that above.
Before our Father's throneWe pour our ardent prayers;Our fears, our hopes, our aims are one,Our comforts and our cares.
We share our mutual woes,Our mutual burdens bear,And often for each other flowsThe sympathizing tear.
When here our pathways part,We suffer bitter pain;Yet, one in Christ and one in heart,We hope to meet again.
From sorrow, toil, and pain,And sin we shall be freeAnd perfect love and friendship reignThrough all eternity.
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All of the cited passages deal with unity or one-ness. This is also the theme of the hymn, but there's very little in the hymn text that bears any strong resemblance to these passages. The clearest similarity I can find is between the lines "The fellowship of kindred minds / Is like to that above" and the passages from John 17 ("'that they may be one, even as we are one.'").