Washington (CNN) An influential Republican senator is urging President Donald Trump to up the ante in talks over his border wall: To push for an increase in the national debt limit as part of a legislative package to avert another government shutdown in mid-February.

GOP Sen. Lindsey Graham, a close ally of the President, told CNN that he spoke with Trump at dinner Monday night about "what would a good deal look like" as lawmakers attempt to negotiate an agreement to stave off another government shutdown -- and floated the possibility that raising the debt limit could be part of the negotiation.

Graham, the senior senator from South Carolina, said he is "hopeful we can solve more than one problem" and added "I think the President understands we need to raise the debt ceiling. It comes due in March, so why not just expedite things."

Taking such a step would amount to a dramatic move by the President: He could dare Democrats to block funding for the border wall and risk sending the country into default in the process. Graham said agreeing to one big package could clear the decks of the thorny issues facing Congress in the weeks ahead.

Congressional negotiators are racing against the clock to strike a deal that would avert another shutdown after the President signed into law a stopgap funding bill last week that put an end to the longest federal government shutdown in US history. The shutdown was triggered by a standoff between congressional Democrats and the President over Trump's demand for $5.7 billion for a border wall, which Democrats have refused to agree to.

Read More