The Reds' vice-captain has not been seen in a Liverpool shirt since sustaining the issue in the second half of a 3-0 Premier League victory over Aston Villa at the start of September.
Alexander-Arnold was forced to pull out of the England squad for their games against Ukraine and Scotland, while he has also missed Liverpool's clashes with Wolverhampton Wanderers, LASK Linz and West Ham United over the past couple of weeks.
However, the 24-year-old managed to avoid a serious injury, and Jurgen Klopp's assistant Pep Lijnders has revealed to liverpoolfc.com that Alexander-Arnold is now taking part in team exercises at their Kirkby training base.
Lijnders was unable to confirm Alexander-Arnold's availability for Wednesday's EFL Cup third-round tie with Leicester City, though, saying: "Trent trained yesterday with us but it was just in the rondo and the ball possession with the counter-pressing, but he was outstanding. You see him instantly, of course. He is with the team training, I am not sure if he will be ready, we will see today.
Without Alexander-Arnold in the ranks, and with Calvin Ramsay plying his trade on loan with Preston North End, Joe Gomez would be the natural pick to fill in at right-back against the Foxes.
The Englishman has taken advantage of Alexander-Arnold's injury and the recent absences of Virgil van Dijk and Ibrahima Konate - through suspension and a thigh problem respectively - to play the full 90 in each of Liverpool's last three Premier League games.
Gomez was given a torrid time by Pedro Neto in the Reds' 3-1 win at Wolves two weekends ago, though, and Stefan Bajcetic - just back from injury himself - was trialled in an Alexander-Arnold-esque inverted role against LASK Linz in the Europa League.
Bajcetic is expected to be involved on Wednesday after being rested against West Ham, but Lijnders singled out Gomez for praise, claiming that the 26-year-old has showcased "incredible" performances in Alexander-Arnold's absence.
"But then if we speak about Trent we have to make a compliment to Joe Gomez because the way he played the role, played the role in his way but still being able to come inside as well, was just incredible," Lijnders added.
"His level and attitude, just a big compliment to him because you are replacing one of the best full-backs in the world. He did really, really well. That's good and shows a lot about our squad as well, that we can have that."
Aside from Alexander-Arnold, the Reds' only other injury concern is experienced midfielder Thiago Alcantara, who has been on the treatment table since April with a troublesome hip problem.
Thiago recently suffered a setback in his rehabilitation, and while Lijnders confirmed that the Spaniard is in a "good way", it will seemingly still be a while before he is back in competitive action.
"He is in a good way, the healing process is going well so that's the most important [thing] at this moment," the 40-year-old added when asked about Thiago.
Liverpool enter this year's EFL Cup aiming to win a record-extending 10th title, but the defence of their 2021-22 crown ended in the fourth round last year with a 3-2 loss at Manchester City.