The English striker, who last featured for Spurs on 23 August against Manchester City, had been attempting a return to full fitness but suffered a setback last week.
Solanke had initially returned to training earlier this month, raising hopes that he could feature in upcoming fixtures, including Tottenham's Champions League away clash against Bodo/Glimt on Tuesday.
However, he missed training again on Monday ahead of the European tie, prompting the decision to undergo surgery.
"Dom has got that ankle issue that's been bothering him for a little while, so now we've decided to make a minor surgery," Frank told a press conference.
"It's a small procedure, so that will mean he is not ready for today of course and of course Leeds (on Saturday) as well. We'll have more news about a timeframe after the international break, but I don't expect it to be long."
Frank had previously understated concerns during a Friday press conference, even as Solanke's absence continued. Tottenham drew 1-1 at home to Wolves on Saturday night, a match in which the striker was unavailable.
Solanke had already missed much of pre-season due to the ankle problem, but appeared fit in early August, playing against Paris Saint-Germain, Burnley, and Manchester City.
The procedure is aimed at resolving the injury fully and preventing further complications, though Spurs have yet to announce a timeline for Solanke's return.
The club will likely monitor his recovery closely in the coming weeks, as the forward's presence could prove vital in both domestic and European competitions.
Manchester City secured a second successive Premier League victory at home against Sunderland, Chelsea were held scoreless away to Bournemouth while Tottenham managed to beat Brentford.
Thomas Frank has said he is fully prepared to put emotion aside as Tottenham face Brentford on Saturday, insisting he will celebrate any Spurs goal despite hosting the club where he spent nine transformative years.
When Thomas Frank returns to face Brentford, now as manager of Tottenham Hotspur, the stakes could not be higher.
Tottenham snatched a dramatic 2-2 draw at St James' Park as Cristian Romero produced a 95th-minute overhead kick to rescue a point against Newcastle.
