Whether he'll be able to play Sunday against the Green Bay Packers in London remains in doubt.
"Daniel is feeling a little bit better today," Daboll said. "We'll just see how that goes."
Daboll would not answer a question about whether Jones has a high ankle sprain. He declined to provide more specifics.
The initial tests on Jones' injured ankle have been positive. There is no fracture. Jones was hurt when his left foot got caught underneath safety Jaquan Brisker on a sack in the third quarter of Sunday's 20-12 win over the Chicago Bears.
The Giants' starting quarterback left the game but went back in after backup Tyrod Taylor exited with a concussion. Jones admitted afterward that he was in pain and his ability to move was compromised.
Daboll said Monday that Taylor was in the concussion protocol and would also be monitored throughout the week.
Davis Webb would start Sunday against the Packers if Jones and Taylor are unable to play. Webb is currently on the practice squad.
"In terms of where we're at this point, I don't think we're at that point just yet to make a decision," Daboll said. "Let this thing play out."
Webb was the Giants' third-round pick in the 2017 NFL draft. He spent time with the New York Jets and was with Daboll in Buffalo before re-signing with the Giants this offseason.
Webb has never thrown a pass in an NFL game.
"I think you try to plan for every contingency plan that you may or may not have throughout the week. [Starting Webb] will certainly be one of them," Daboll said. "Again, I don't want to guess right now on Monday where all the other guys are going to be. Davis has been in our offense for quite some time. He knows the ins and outs of it, so if he has to play we'll do everything we can, and I know he will, to be ready to play."
The Giants planned to work out quarterbacks early this week as another contingency. They would need a backup for Webb if Jones and Taylor are unable to play. Daboll still was not certain whether they will add one to roster.
Taylor left Sunday's game after taking a helmet-to-helmet hit early in the fourth quarter as he scrambled for a first down. Daboll said the current climate after the handling of Miami Dolphins quarterback Tua Tagovailoa's injury will not affect the team's treatment of Taylor.
"I think we take every injury serious here. We try to do everything we can do to put the player's best interest in mind. That is what we'll always do," Daboll said.
Jones' injured ankle could prove troubling in a different way. The Giants have leaned heavily on his running skills this season. Jones rushed for a pair of touchdowns before the injury on Sunday. He has 279 yards rushing on 45 carries already this year. His career high is 65 carries.
Jones has missed starts in each of his first three professional seasons. He missed a pair of starts his rookie season with a high ankle sprain.
"I'm just going to take it each day and see how [Jones] is," Daboll said. "If he can practice and go out there and perform the things we need him to do, great. If he needs a day, he needs a day. I think it's case by case with all those guys. He's obviously played a lot of football. I trust him and where he's at. I think we as coaches need to do a good job watching him, evaluating him. Getting all the information we need to get to make the best decision for him and the team as we can."