The biblical basis Synergists would claim are all texts that speak of individuals agreeing, or choosing, to put their faith in Christ for salvation. Perhaps the most representative of those would be Revelation 1:20 which has Christ say:
"Here I am! I stand at the door and knock. If anyone hears my voice
and opens the door, I will come in and eat with him, and he with me."
Jesus, however, is not addressing unbelievers in need of salvation, but Christians who are already saved, yet who don't have Christ present at their spiritual 'table'. Now, I don't wish to misrepresent Synergists, so if that is not a text they would use, then I hope they will correct me.
They could stress Deuteronomy 30:15 & 19 where a choice is put before the nation of Israel, whether to choose God's ways and so to choose life, or to follow the ways of the nations. Another similar text is Joshua 24:15 where the nation is challenged to choose for themselves, that day, who they would serve. To really answer the question, just consult a Bible Concordance under "choose" to find which ones speak of the unsaved choosing to be saved, or choosing not to be saved. Note that verses such as Proverbs 1:29 speak of not choosing to fear the Lord. Others speak of choosing the right and rejecting the wrong, but Synergists would have to argue their case for that applying to the act of being saved.
'Synergism' means 'working together', while monergism means "one work". Synergists believe that two forces are required to bring about the saving of a soul - the Holy Spirit and the will of man. If a person resists the Holy Spirit, then there will be no 'working together' to achieve that person's salvation. Synergists could call upon Philippians 2:12-13 for that (though 1:5-6 is part of that equation, meaning the theology in those four verses might not speak of the act of being saved initially.)
It would be better if Synergists answered your question, really, but given that that has not yet happened, hopefully this will get the ball rolling for you.