Scalise was shot on June 14 during a congressional baseball practice in Alexandria, Virginia, by James Hodgkinson, a 66-year-old of Belleville, Illinois.
The officials said investigators don't know what the list of names means, but members of Congress have been briefed. This person was not authorised to speak publicly about the case and spoke on condition of anonymity.
The president said that because of the "tremendous pain and suffering he's now enduring. our country will perhaps become closer, more unified".
"On behalf of Steve and our children, I want to thank everyone from the bottom of our hearts for the incredible amount of prayers and warm wishes we have received since Wednesday's events", Scalise's wife, Jennifer, said in a statement Friday.
Scalise was shot in the hip, and the bullet damaged bone, blood vessels and internal organs.
A spokesman for George Washington University Hospital said that Matt Mika, a lobbyist wounded in the attack, was improving, and that as of Thursday night his condition had been upgraded from critical to serious.
"Congressman Steve Scalise is in serious condition", said the notice from the whip's office.
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But efforts to persuade North Korea to release the men now held had not been successful until Warmbier's release Tuesday. Asked whether he was working on the release of detained Americans , he said, "that is not my objective right now".
Scalise has undergone multiple surgeries and he is expected to need more surgeries.
The gunman was shot and killed by Capitol Police officers on scene, but not before one of them, Special Agent Crystal Griner, suffered a gunshot to his ankle.
Several other people were also injured in Wednesday's shooting before Scalise's security detail and other police officers gunned down the assailant, who later died. Authorities were investigating Hodgkinson's social media posts, a cellphone, computer and camera.
Nonetheless, Sava said, "We fully expect him to be able to walk" and "hopefully run".
Trump added that Scalise is "having a hard time, far worse than anybody thought" following the shooting Wednesday.
The surgeon on Friday said he was "a lot more confident and optimistic" about Scalise's recovery than he was two days ago.