"Hopefully," he said when asked if he could play Thursday. "I got [Tuesday], I got Wednesday. We'll figure that out when Thursday comes."
Anthony has missed three games -- during which the Knicks have gone 0-3 to fall to 9-21 -- but he ran on Sunday and Monday for the first time since injuring the ankle Dec. 23 against Orlando.
"Getting there, feeling a lot better," he said, adding that he hasn't needed any additional testing. "Each day is getting better and better. Still some things lingering but each day is getting better."
Asked if the ankle injury was worse than he originally thought, Anthony, who had hoped to return for the Christmas Day game against Oklahoma City, said: "Yeah, it was worse than I thought. The night I sprained it, I was trying to get back in the game. I went back and taped it up. As soon as I walked out of the locker room, I couldn't do it. Then the next day it just swelled up on me big time. It kept swelling, kept swelling so we had to get control of that first and then go from there."
When he does return Anthony said he will "have to play with a little bit of tenderness, a little bit of pain."
As for the Knicks' dreadful start, Anthony doesn't seem to have any easy answers and said "effort" continues to be an issue.
"The effort is definitely not there some days," he said. "Or it's there and then sometimes it's not. We play with effort in spurts so we gotta get consistency from that standpoint."
Anthony has said he will opt out after this season and test free agency, although he's also on record saying he wants to retire in New York.
If the Knicks continue to lose and fail to make the playoffs, it's unclear how much that would impact his future plans.
For now, he has no solutions to the team's struggles.
"We can't change [our record] at this point so we gotta...This is not how I envisioned it, this is not how we envisioned it coming into this season," he said. "But it is what it is at this point and we can't be crying about it. We gotta just find a solution to it. We gotta take it one game at a time and figure it out."
Pressed for answers to the team's problems, Anthony said: "I really don't know what's gonna happen. I can't be too concerned about that. I just gotta worry about playing ball and whatever happens, happens."
Anthony did say he backs coach Mike Woodson, who has already received a vote of confidence from owner Jim Dolan.
"Coach Woodson is a strong person, a strong-willed person," Anthony said. "His message is definitely getting through to people, especially through to me and guys who have been here and understand the situation.
"We understand it. It's so easy to just fold in this moment right now, so his message is very clear throughout this locker room."
Adam Zagoria covers the New York Knicks and Big East hoops for NBA.com and SNY.tv. He also appears as a Big East Basketball Insider on SNY and 1050 ESPN Radio. You can follow him on Twitter and read his blog.