The Senate voted along party lines on Wednesday to take up Republicans’ $70 billion immigration crackdown bill, but several G.O.P. senators suggested they would not allow it to pass without new language barring President Trump from creating a fund to compensate his political allies.

The 53-to-46 vote began debate on legislation that was supposed to unite Republicans around Mr. Trump’s hard-line immigration enforcement agenda, but has instead become a flashpoint for clashes between the president and G.O.P. members of Congress just months before the midterm elections.

Before bringing it up, Republicans followed through on a decision they made late last month to jettison $1 billion in security funds for Mr. Trump’s ballroom and other White House renovations, which had drawn a mini-revolt within his party.

They were also discussing adding a provision that would block Mr. Trump from resurrecting the idea of creating a $1.8 billion fund to pay people who claim to have been victimized by the government, which the administration dropped this week after it came under withering attack from members of both parties.

The vote set the stage for a lengthy vote-a-thon on a range of issues that is expected to culminate on Thursday in Senate passage of the immigration enforcement bill, which Republicans are pushing through Congress using special procedures that shield it from a filibuster. To do so, they will need near unanimity to overcome united Democratic opposition.

Even after sworn testimony on Tuesday by Todd Blanche, the acting attorney general, that the administration was not now — or ever — moving forward with the fund, a significant bloc of Republican senators still wanted to codify that statement in the bill. Some of them said their votes for the measure, which funds Immigration and Customs Enforcement and Customs and Border Protection, were contingent on doing so.