WASHINGTON—President Trump is leaning toward Emmet Flood as the next White House counsel, but West Wing officials have urged Mr. Trump not to rush the pick, according to people familiar with the matter.

Mr. Trump told associates over the weekend that he wants Mr. Flood, currently serving as White House special counsel, to succeed Don McGahn, these people said. Mr. McGahn is planning to leave his role as top White House attorney after the Senate acts on the nomination of Judge Brett Kavanaugh to the Supreme Court.

Mr....