Trump's comments came as Washington joins intensifying international efforts to heal the worsening rift between the key Western Gulf allies, which has escalated into the region's worst diplomatic crisis in years.
Qatar denies the allegations and has sent its top diplomat on a tour of European capitals in a bid to drum up support.
But its neighbours seized on Trump's remarks as vindication of their position and of the crippling sanctions they imposed on Monday.
The United Arab Emirates welcomed "President Trump's leadership in challenging Qatar's troubling support for extremism".
"The next step is for Qatar to acknowledge these concerns and commit to reexamine its regional policies," its ambassador to the United States Yousef al-Otaiba told the official WAM news agency.
"This will provide the necessary basis for any discussions," he added.
Saudi Arabia said an immediate change of policy by Qatar was essential.
Bahrain "stressed the necessity of Qatar's commitment to correct its policies and to engage in a transparent manner in counter-terrorism efforts", its official BNA news agency said.
The three governments made no comment on separate remarks by US Secretary of State Rex Tillerson urging them to ease their land and sea blockade of Qatar, which he said was hindering the US-led campaign against ISIS and having humanitarian consequences for ordinary people.
Qatar's neighbours have given its citizens 14 days to leave, banned Qatari flights from their airspace and closed its only land border.
Qatari Foreign Minister Sheikh Mohammed bin Abdulrahman Al-Thani was in Moscow on Saturday after talks in Germany on Friday.
Qatar ally Turkey meanwhile hosted the Bahraini foreign minister for talks on the crisis.