Washington(CNN) A Democratic lawmaker on Thursday ratcheted up warnings to the Trump administration amid a growing standoff over subpoenas and oversight requests the White House says it will resist.

Rep. Gerry Connolly threatened jail time for White House officials who are declining to comply with congressional committees' efforts to conduct oversight of President Donald Trump's administration. It's the latest salvo in the escalating battle between the White House and congressional Democrats, who have scaled up their oversight requests following the conclusion of special counsel Robert Mueller's investigation.

Connolly, who sits on the Oversight Committee, told CNN's Wolf Blitzer on "The Situation Room," "We're going to resist, and if a subpoena is issued and you're told you must testify, we will back that up."

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"And we will use any and all power in our command to make sure it's backed up -- whether that's a contempt citation, whether that's going to court and getting that citation enforced, whether it's fines, whether it's possible incarceration," the Virginia Democrat added. "We will go to the max to enforce the constitutional role of the legislative branch of government."

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Trump and administration officials have signaled they will resist the attempts to investigate his policies and personal finances, which has outraged lawmakers and led to accusations he is undermining the constitutional separation of powers.

READ: White House letter that says Stephen Miller won't testify before Congress on immigration

"We're fighting all the subpoenas," Trump said on Wednesday. "These aren't like impartial people. The Democrats are trying to win 2020. They're not going to win with the people I see and they're not going to win against me."

Three officials have refused to comply with congressional requests to testify at the administration's urging, ratcheting up the long-brewing battle with House Democrats ready and willing to take their multi-front investigations to the courts.

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The House Oversight Committee had asked to hear from senior White House adviser Stephen Miller on immigration, former security official Carl Kline on security clearances and Justice Department Principal Deputy Assistant Attorney General John Gore on adding a citizenship question to the US census.

Connolly called the officials' excuses for failing to appear "an assault on the legislative branch" and "the constitutional framework of our government," warning that the legislative branch would become "a pale shadow of what it was intended to be" should the officials' refusals stand.

Connolly said his committee had hoped to learn what Miller "has in his head" given his increasing involvement in the administration's immigration policies, calling him Trump's "immigration whisperer" in light of the Department of Homeland Security's recent leadership changes and the "zero tolerance" policy that resulted in family separations.

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"We want to hear from him, what is your thinking, what is it you've been advising the President, and where is it you think you're going to be taking us as a country with these kinds of policies and personnel changes?" he added.