President Donald Trump should be cautious of newly found support among moderate Republicans and Democrats, former White House chief of staff John Sununu said Monday.

"It's always important to stay close to the people that brought you to where you are. New found friends from the opposition are, at the very best, short term. And at the very worst, long term, and undermine you behind the scenes," Sununu, who was chief of staff under former President George H.W. Bush, said on "Squawk Box."

Some Democrats and moderate Republicans offered cautious praise of Trump after he ordered a military strike against a Syrian air base Thursday night. The attack was in response to Tuesday's deadly chemical weapons attack allegedly carried out by Bashar Assad's military on a rebel-held area.

As Trump turned to the Syria crisis, a Wall Street Journal report said he is also considering a major shake-up of his staff, including whether to keep his chief of staff, Reince Priebus, and his chief strategist, Steve Bannon — a claim which the White House denied.

Numerous reports said Bannon, who joined Trump's campaign after serving as chairman of Breitbart, a news website, was considering leaving the White House after his removal from the National Security Council's principals committee.

The White House has been under pressure amid Trump's low approval ratings and failure to pass an Obamacare repeal and replacement bill.

Sununu warned on Monday that it would not be a good time for a major White House change to take place.

"Intrigue in the White House is always exaggerated," Sununu said. "They've got to get momentum and you can't have new people coming in and trying to restart this thing."