Sen. Lindsey Graham says he encouraged Trump during a telephone conversation Sunday to reopen government for a short period to try to negotiate a deal with Democrats to end the impasse.
But the South Carolina Republican tells "Fox News Sunday" that Trump wants a deal first.
House Speaker Nancy Pelosi, a California Democrat, wants Trump to end the shutdown first.
Graham says he thinks Trump is willing to accept the level of wall funding he is seeking, along with some immigration measures Democrats might accept, such as helping immigrants who were illegally brought to the U.S. as minors.
Trump has said that while he wants a broader immigration deal, he first wants to hear what the Supreme Court has to say about the group known as "Dreamers."
President Donald Trump is pushing back against the idea that he doesn't have a strategy to end the partial government shutdown, now into its fourth week.
At the same time he's not saying what his strategy is.
Trump says in a Fox News Channel interview that he has "no idea" whether he can get a deal with House Speaker Nancy Pelosi. The California Democrat opposes spending money on what she calls an "ineffective, wasteful wall" along the U.S. southern border.
Funding for the border wall remains the sticking point in passing budgets to reopen the government.
Trump continues to hold out the possibility that he'll declare a national emergency to build the wall without congressional approval.