Republican leaders acknowledge they will need to persuade their most conservative colleagues — including several potential presidential contenders — to be satisfied with legislative gains that might fall short of their ideals. Insisting on all their demands could leave Republicans short of 51 votes, let alone the 60 that will still be required to pass most bills.

“We are going to have to convince people that we are not going to be perfect, but let’s at least move the ball down the field and try to do things many of us have wanted to do for a long time,” said Senator John Thune of South Dakota, chairman of the Senate Republican Conference.

While they try to entice colleagues on the right, Senate Republican leaders must be mindful of the potential vulnerability of their senators up for re-election in 2016 — when the electoral climate will almost certainly be less favorable to them — in presidential swing states such as Illinois, Pennsylvania and Wisconsin. They also would be contending with a Democratic minority eager to exact revenge for the difficulties Republicans caused them over the previous eight years.