The report said Netanyahu prefers to delay any action until after his meeting with Trump, citing an unnamed senior Israeli source. Netanyahu will begin a visit to the US on Sunday.
The decision comes as negotiations on the second phase of the deal set to begin Monday.
Israel has not publicly reacted to the reports.
Additionally, Netanyahu reportedly canceled a planned meeting with Mossad chief David Barnea, Shin Bet chief Ronen Bar and other senior negotiators, instead sending his military secretary, Roman Gofman, to inform them of the delay.
The Israeli leader is also said to be considering appointing Strategic Affairs Minister Ron Dermer as head of the team, arguing that the talks are more diplomatic than security-focused, the Times of Israel news outlet reported.
The ceasefire and prisoner exchange agreement between Hamas and Israel, which took effect Jan. 19, has resulted in the release of 583 Palestinian prisoners, 13 Israelis and five Thai nationals. The first phase of the deal aims to release more than 1,900 Palestinian prisoners and 33 Israeli captives within 42 days.
Israel announced that talks on the second phase of the ceasefire and prisoner swap deal would begin Feb. 3 in Washington, DC.