As a federal investigation into Rudolph W. Giuliani escalates, his advisers have been pressing aides to former President Donald J. Trump to reach into a $250 million war chest to pay Mr. Giuliani for his efforts to overturn the results of the 2020 election on Mr. Trump’s behalf. The pressure from Mr. Giuliani’s camp has intensified since F.B.I. agents executed search warrants at Mr. Giuliani’s home and office last week, according to people familiar with the discussions, and comes as Mr. Giuliani has hired new lawyers and is facing his own protracted — and costly — legal battles. Federal prosecutors...