(CNN) Several senior Senate Republicans and members of GOP leadership raised serious concerns Monday about the possibility of President Donald Trump bypassing Congress and using an emergency declaration to build a border wall with Mexico.

Those GOP lawmakers said such a move could have broad constitutional problems, practical political implications for Republicans and could set a dangerous precedent for all future presidents who might use the power to advance their policy priorities despite Congress.

The words of caution come as Trump's demand for billions to build a border wall remains a key point of contention in private negotiations to avert a government shutdown in just 12 days, sources from both parties said Monday.

"There are lot of reservations in the conference about that and I hope they don't go down that path," Senate Majority Whip John Thune of South Dakota told CNN about the emergency declaration that Trump has repeatedly said could be his backstop position if talks don't yield funding for a wall.

Sen. John Cornyn, a Texas Republican, called it a "dangerous step" and "not a good strategy" and that Republicans have conveyed these doubts to the White House "many times. Publicly and privately."

Read More