Israeli military said the Red Cross transferred to its custody the coffins with the remains
The military said it would soon receive the bodies from the Red Cross.
"Hamas is required to uphold the agreement and take the necessary steps to return the hostages," the military said in a statement.
According to the ceasefire agreement, Hamas must hand over a total of 28 bodies. Following the return of four bodies on Monday, 20 now remain in the hands of the militant group.
Hamas has said there are difficulties in locating the remains in the rubble after two years of war in the Gaza Strip.
On Tuesday, Hamas said Israel was killing people in Gaza and violating the ceasefire. Trump suggested Hamas was reneging on its promise to return the dead, and threatened the group with violence.
"If they don't disarm, we will disarm them," Trump told reporters in Washington after returning from his weekend trip to the Middle East. "And it will happen quickly and perhaps violently."
Meanwhile, Hamas swiftly reclaimed the streets of the Gaza Strip's urban areas following the partial withdrawal of Israeli troops last week.
In a video circulated late on Monday, Hamas fighters dragged seven men with hands tied behind their backs into a Gaza City square, forced them to their knees and shot them from behind as dozens of onlookers watched from nearby shopfronts.
A Hamas source confirmed that the video was filmed on Monday and that Hamas fighters participated in the executions.
Reuters was able to confirm the location by visible geographic features.
Trump has given his blessing to Hamas to reassert some control of the strip, at least temporarily.
Israeli officials, who say any final settlement must permanently disarm Hamas, have so far refrained from commenting publicly on the re-emergence of the group's fighters.
On Monday the US president proclaimed the "historic dawn of a new Middle East" to Israel's parliament as Israel and Hamas were exchanging the last 20 living Israeli hostages in the Gaza Strip for nearly 2000 Palestinian detainees and prisoners.
While Israeli officials had understood there could be some delay in the recovery effort, the families of hostages and their supporters expressed dismay that only four bodies were returned on Monday.
The Hostages Family Forum, a grassroots organisation representing many of the hostage families, called it a "blatant violation of the agreement by Hamas".
The top official in Israel co-ordinating the return of hostages and the missing, Gal Hirsch, told the families in a note that pressure was being applied on Hamas through international mediators to expedite the process.
Aid trucks have yet to be permitted to enter the Gaza Strip at the full anticipated rate of hundreds per day and plans have yet to be implemented to open the crossing to Egypt to let some Gazans out, initially to relocate the wounded for medical treatment.
Meanwhile, Israeli drone fire killed five people as they went to check on houses in a suburb east of Gaza City and an air strike killed one person and injured another near Khan Younis, Gazan health authorities said.
The Israeli military said it had fired on people who crossed truce lines and approached its forces after ignoring calls to turn back.
Hamas sources told Reuters on Tuesday the group would tolerate no more violations of order and would target collaborators, armed looters and drug dealers.
with AP and DPA
