He has worked for decades as a partner with the committee's senior Republicans.
That, in turn, has led to criticism from senior Republicans.
A final decision on what do about immigration policy awaits a meeting this week of senior Republicans.
At the time of his retirement, he was the most senior Republican in the Senate.
Some of those senior Republicans are, in fact, at the center of the effort to find a compromise.
Even with this push from the two senior Republicans in Congress, the bill never came up for a vote.
Some senior Republicans said the chances of either proposal now making it into law had dimmed.
He is the most senior Republican in the Senate.
But senior Republicans who are familiar with the conversation said the Speaker left no doubt that he would have taken such a step.
"George has won the nomination largely because he managed to identify himself with the President," a senior Republican said last week.