Several GOP officials said Thursday that Republican leaders discussed the possibility privately this week, recognizing that unified Democratic opposition to the accounts has so far stalled efforts to advance the president's top domestic priority.
At the same time, these officials said GOP leaders were wary of leaving the impression they intend to abandon Mr. Bush's proposal to allow younger workers to invest a portion of their payroll taxes independently.
In the end, leaders remain determined to deliver what the president wants, they added.
"The attempt ... has always been to find Democrats to come on board with a bipartisan plan, and what has been stopping this is the issue of personal accounts," said one official familiar with the deliberations.
Any shift in tactics has the potential to embolden Democratic critics who say Mr. Bush's proposal would privatize Social Security and lead to benefit cuts and a large increase in the national debt.
At the same time, it could undercut the current 60-day campaign by the White House to build public support for the president's approach.
White House spokesman Trent Duffy stopped well short of endorsing the move.
Mr. Bush "wants to save Social Security permanently and give future workers personal accounts and a better retirement," Duffy said. "He welcomes any activity that moves us in that direction."
A spokeswoman for Senate Democratic leader Harry Reid declined comment.
Reid has said numerous times he is willing to seek bipartisan legislation on Social Security once the president abandons his call for personal accounts.
Republican officials said senior Senate GOP lawmakers had discussed at least two approaches, without a clear decision on how to proceed.
One involved quietly acquiescing in an effort by one of several senators with an interest in Social Security to see whether they could attract Democratic support for a bill that addressed the program's solvency problems but omitted personal accounts.
The other involved sanctioning an effort by Sen. Charles Grassley, chairman of the Senate Finance Committee, to see whether Democrats on the panel would be willing to join in talks.