Alexander-Arnold was forced to pull out of England duty after suffering a hamstring injury, and on Tuesday morning, the right-back shared an update from the NAS Sports Complex.
However, with Milner also not in action until next month having retired from England in 2016, the midfielder has opted for a more low-key international break so far.
In a post on Instagram on Tuesday, the 36-year-old showed himself “back at it” in the gym at the AXA Training Centre, along with the hashtags #fullbodyday and #readyfortherunin.
Milner missed the last two games for Liverpool after testing positive for COVID-19, but will be back in contention for the visit of Watford on April 2.
His return to availability could be particularly useful if Alexander-Arnold is out for the first two games next month as expected, with the veteran able to fill in at right-back.
“He just has to be available and then that is fine.”
Whether Milner will revert to first-choice stand-in remains to be seen, though, with Joe Gomez impressing at right-back in the 1-0 victory at Forest and also able to use the international break to train and recover.
But it certainly boosts Klopp’s options ahead of a busy April that could see Liverpool play up to nine times, including two Champions League quarter-final legs against Benfica and a pair of clashes with Man City.