Clarke wants the 12 play-off teams to lobby FIFA and stop bookings picked up in qualifying matches from counting in the March knockout games.
Scotland and Wales are both at real risk of losing players should they progress to a play-off final.
Page said of Clarke’s call: “I fully support it. Absolutely.
“I see it especially from Scotland’s point of view because they had six teams in their group and had more games to play.
“It is what it is. But if we can influence a change we absolutely will.”
Scotland have eight players – Andy Robertson, Billy Gilmour, Che Adams, Jack Hendry, John McGinn, Kevin Nisbet, Scott McTominay and Stephen O’Donnell – one booking away from missing the play-off final were they to get there.
Wales will be without Joe Morrell for their home semi-final after the midfielder picked up a second caution against Belgium on Tuesday.
Seven others will be walking a disciplinary tightrope in March – Aaron Ramsey Chris Gunter, Harry Wilson, James Lawrence, Joe Allen, Kieffer Moore and Sorba Thomas.
Ramsey and Ben Davies missed the Euro 2016 semi-final defeat to Portugal through suspension after receiving second yellow cards in the tournament.
That rule was changed ahead of Euro 2020, when single cautions from matches expired after the quarter-finals.
Page said: “It’s so frustrating. In Euro 2016 you had influential players missing the most important stage of that competition. They have addressed that one.
“In a couple of years’ time they might do it. Whether or not we have the power or clout to do it and ask for change now and be listened to, I’m not too sure.”
Wales captain Gareth Bale faces a spell on the sidelines at Real Madrid after missing the 1-1 draw with Belgium through injury.
Other key players, such as Juventus midfielder Ramsey, Liverpool full-back Neco Williams and Tottenham centre-back Joe Rodon, have been short of game-time at their respective clubs.
Page has told his players that first-team football is vital for their Wales careers and will be keeping a close eye on the January transfer window.
“Players will want to play games,” Page said. “For whatever reason domestically they don’t, that is fine, every manager has his own opinions.
“I can’t influence that. All I can influence is encouraging the players to want to go and play domestic football, whether it is at their parent club or elsewhere.
“That is what we are encouraging them to do.
“Certainly when you have got all the players we have had for this camp fit and the creativity we have in the team and the goals we can score, you want them all playing to be in the best possible shape for Wales.
“It will be no different in March.”