US President Donald Trump and House Speaker Nancy Pelosi are playing a game of political brinkmanship, seeing who will blink first, political observers say.

A day after Pelosi proposed that Trump postpone his State of the Union speech until the government reopens, the president told her in a letter on Thursday that she couldn't use military aircraft for her planned overseas trip.

The latest dust-up between what many see as the two most powerful politicians in Washington suggests they're digging in on the partial government shutdown, experts said.

In a letter to the speaker, released by the White House on Thursday, Trump wrote: "Due to the shutdown, I am sorry to inform you that your trip to Brussels, Egypt and Afghanistan has been postponed. We will reschedule this seven-day excursion when the shutdown is over.

"In light of the 800,000 great American workers not receiving pay, I am sure you would agree that postponing this public relations event is totally appropriate. I also feel that, during this period, it would be better if you were in Washington negotiating with me."

Pelosi and other lawmakers were planning to fly to Afghanistan on Thursday afternoon, a trip that would include "a required stop" in Brussels for pilot rest, where the delegation was scheduled to meet top NATO commanders, US military leaders and key allies, her spokesman, Drew Hammill, said in a statement.

"This weekend visit to Afghanistan did not include a stop in Egypt," Hammill said, noting a difference in Trump's description of Pelosi's trip.

Trump's letter came a day after Pelosi wrote to the president suggesting he postpone his Jan 29 State of the Union address "given the security concerns and unless government reopens this week", amid a stalemate that has shuttered large parts of the federal government since Dec 22, the longest closure in US history.

The president and congressional Democrats are wrangling over $5.7 billion for Trump's signature wall along the US-Mexico border, one that Trump had pledged would be paid for by Mexico.

Democrats have offered $1.3 billion as their cap for border barriers and fences. Continued efforts from both sides of the aisle in Congress so far have fizzled, including the latest one on Wednesday, when Trump hosted a bipartisan group of lawmakers from the Problem Solvers Caucus to discuss a way forward.

Many lawmakers want to reopen the government and negotiate the wall money afterward. Trump has dismissed that idea, according to a report by thehill.com.

Cal Jillson, a political scientist and historian at Southern Methodist University in Dallas, Texas, said that what happened over the past few days between the president and the speaker is like a "schoolyard tit-for-tat".

"You want to cancel my State of the Union message, so I'll cancel your overseas trip. This fairly childish back-and-forth does not suggest that a solution to the shutdown is likely anytime soon," he said in an email.

Pelosi's goal in suggesting that the annual State of the Union address be postponed is to highlight the disruption caused by the partial government shutdown and to disrupt, if not deny, the president the opportunity to blame Democrats for it in the speech, Jillson said.

"If the Democrats blink first, they will be forced to give Trump money for his wall, while if Trump blinks first, he will go away empty-handed or will, at most, get money for border security broadly but no money for a wall," he said.

If that is the case, Trump would have the most to lose because he made "a big, beautiful wall" a centerpiece of his 2016 presidential campaign and has been unable to deliver it to date, Jillson said.

"I expect the Trump administration to find a way to open the government without seeming to cave completely because they know they are taking most of the blame for this increasingly untenable situation," he said.

Jillson said that in democratic politics, compromise, accepting half a loaf, allowing both sides to declare victory, is the way such matters usually conclude.

"In this case, providing Trump $5 billion for border security, with $1.5 billion to $2 billion for a physical barrier, and giving the Democrats a DACA fix has been under discussion," he said, referring to Deferred Action for Childhood Arrivals, a program that allowed deportation relief for immigrants brought to the country illegally as children, for which some Senate Republicans have proposed a reauthorization.

"Perhaps something this sensible may be where we will end up," Jillson said.

Stanley Renshon, a political scientist at the City University of New York, said Pelosi's proposal on the high-profile State of the Union is "one more shot in a raging, larger battle" between the Democrats and their allies on one side, and Trump and his supporters on the other.

"My guess is that it is going to go on for quite a bit longer because it does matter who blinks first," Renshon said.

Renshon argued that Trump is playing things "very craftily" in terms of mitigating the impact of the shutdown.

"He's doing some smart things. One of the things he's doing that's smart is he's renegotiating the group of people who are required to come in to work," Renshon said.

Among them are workers at the Internal Revenue Service, who process tax returns and refunds, Renshon said.