(CNN) A veteran Senate Republican has some advice for President Donald Trump: Stay out of the next round of border security talks.

Sen. Lamar Alexander, a close adviser to Senate Majority Leader Mitch McConnell, told CNN that the involvement of both Trump and House Speaker Nancy Pelosi would complicate efforts to get a border security deal before a mid-February deadline to avert another government shutdown. And he called for any House-Senate agreement on border security to include legislation preventing shutdowns from happening in the future -- an idea picking up bipartisan steam.

"We know the President's positions. And we know Speaker Pelosi's positions. Now, let them step back and let the conference committee do what it usually does," Alexander said, pointing to efforts to fund physical barriers in the past.

Alexander's comments reflect the growing unease among Senate Republicans over the prospects of another shutdown, which Trump raised Sunday as a possibility during an interview with the Wall Street Journal

McConnell told CNN later on Monday that a shutdown is a "bad idea" when we asked if he could rule it out in the coming round of talks.

Read More