Rudolph W. Giuliani, President Trump’s lawyer who has led the most extensive efforts to damage his client’s political rivals and undermine the election results, discussed with the president as recently as last week the possibility of receiving a pre-emptive pardon before Mr. Trump leaves office, according to two people told of the discussion.

It was not clear who raised the topic. The men have also talked previously about a pardon for Mr. Giuliani, according to the people. Mr. Trump has not indicated what he will do, one of the people said.

Mr. Giuliani’s potential criminal exposure is unclear. He was under investigation as recently as last summer by federal prosecutors in Manhattan for his business dealings in Ukraine and his role in ousting the American ambassador there, a plot that was at the heart of the impeachment of Mr. Trump.