In recent days Trump has repeated that he wants to gain control of the mineral-rich Arctic island.
On Saturday, the US sparked international consternation by sending in special forces to seize Venezuelan President Nicolas Maduro, prompting questions about whether he planned further such actions.
"You will see war powers resolutions introduced on Cuba, Mexico, Colombia, Nigeria, Greenland," Senator Tim Kaine, a Virginia Democratic senator who has led several resolutions seeking to check US presidents' use of military force without congressional approval.
"I just don't know who's necessarily going to be leading all of them, because I will likely be at least a co-sponsor and probably leading some," Kaine told reporters at a news conference ahead of a Senate vote expected on Thursday on whether to halt further military action in Venezuela without lawmakers' approval.
Republican Senator Rand Paul of Kentucky, a co-sponsor of the Venezuela resolution, said he would "keep an open mind" on whether to support any such resolutions.
Paul added that he had not heard any of his fellow Republicans express support for military action to take Greenland.
He called such threats unproductive if the administration wanted to buy the mineral-rich Arctic island.
"If I wanted to buy Greenland, I'd be doing PR events up there, I'd be showering them with gifts, I would be talking to the people of Greenland about how they could be a part of the United States and have their independence. That there'd be great benefits to be under our defence umbrella, and also economic benefits," Paul said.
US Secretary of State Marco Rubio told reporters at the Senate earlier on Wednesday he would meet leaders of Denmark next week to discuss Greenland, but signalled no retreat from Trump's aim to take over the island.
Alarmed allies including France and Germany were working on a plan on how to respond.
Citing the Constitution's requirement that only Congress, not the president, has the right to declare war, Kaine, Paul and other lawmakers have tried and failed repeatedly in recent months to pass resolutions that would have prevented Trump from attacking Venezuela without congressional authorisation.
European leaders affirm Greenland's autonomy
Leaders from major European powers and Canada have rallied behind Greenland this week, saying it belongs to its people.
Britain said its Prime Minister Sir Keir Starmer, who has said he stands with Denmark over Greenland, spoke with Trump on Wednesday and set out his position.
French Foreign Minister Jean-Noel Barrot said he would discuss Greenland with his German and Polish counterparts on Wednesday.
"We want to take action, but we want to do so together with our European partners," he told France Inter radio.
A German government source said Germany was "closely working together with other European countries and Denmark on the next steps".
A senior European official, who spoke on condition of anonymity, said Denmark must lead efforts to coordinate a response, but "the Danes have yet to communicate to their European allies what kind of concrete support they wish to receive".
Johannes Koskinen, chair of the Foreign Affairs Committee of Finland's parliament, called for the issue to be raised at NATO, whose decision-making North Atlantic Council meets next on Thursday.
Koskinen said NATO allies should "address whether something needs to be done and whether the United States should be brought into line in the sense that it cannot disregard jointly agreed plans in order to pursue its own power ambitions."
The world's largest island but with a population of just 57,000, Greenland is not an independent member of NATO but is covered by Denmark's membership.
