The Evening Chronicle says Carroll took part in half a training session with his Newcastle United team-mates for the first time on Tuesday while Matt Ritchie is back running - but Dwight Gayle's calf injury has not improved.
The Magpies have been keen not to put a timescale on a potential return date for Carroll, who has not played since February, after he underwent ankle surgery earlier this year.
Encouragingly, Carroll's ankle has not had an adverse reaction to the physical work he has done since his deadline day move last month and Tuesday marked another step forward in his rehab as he joined in with his team-mates for part of the session.
"The major ones we're talking about - Dwight, Andy Carroll and Saint-Maximin - they're a big three to get fit," Bruce said. "I can't see any of them being available for this weekend against Liverpool but two of them are progressing."