President Trump’s advisers are pulling him in opposite directions over the partial government shutdown, with White House aides pushing to make a deal and political advisers urging him to keep fighting—and use a mail-a-brick campaign against Democrats as a re-election fundraiser.

White House aides are setting up meetings with moderate Democrats and aiming to prepare Mr. Trump to accept a deal for something less than the $5.7 billion he has been seeking for a border wall in exchange for ending the record shutdown, which began...