President Donald Trump has called for the jailing of Illinois officials, accusing them of not doing enough to ensure the safety of federal immigration officers who are conducting raids in Chicago.

The president wrote in a social media post that Chicago Mayor Brandon Johnson and Illinois Governor JB Pritzker, both Democrats, "should be in jail for failing to protect ICE officers!"

Trump has previously called the city a "war zone" amid protests against immigration enforcement. Pritzker has called it an "authoritarian march".

The Republican president spoke as hundreds of National Guard troops arrived in the city as part of his immigration crackdown.

The standoff raises the temperature in an already heated rhetorical battle between Trump and local officials in Democratic-led cities and states.

The arrival of National Guard troops in Chicago follows similar deployments to Los Angeles and Washington DC.

He has also directed them to enter Memphis, Tennessee, and Portland, Oregon. A judge has blocked their deployment to Portland, but allowed them to go to Chicago for now.

The state of Illinois and the city of Chicago filed a lawsuit on Monday to block the deployment.

In a statement, Pritzker called it "Trump's invasion", and said there was "no reason" to send troops into any state without the "knowledge, consent, or co-operation" of local officials.

Trump has threatened to invoke the Insurrection Act and use the military if courts stop the deployments, which he has defended as necessary to address rampant crime and defend immigration enforcers.

In Chicago, protests grew violent last weekend, with immigration authorities saying they opened fire on an armed woman after she and others allegedly rammed their cars into law enforcement vehicles. The woman's condition is unclear, but officials said she drove herself to the hospital.