Both players have struggled to find playing time under David Moyes this season, and Fabio has already started training with Cardiff even though the deal is yet to be completed.
Moyes says "we are delighted for Fabio because he has been a great servant." Fabio's twin brother Rafael also plays for United.
Zaha joined United from Crystal Palace this season but has not started a single Premier League game. Moyes says "we ... just felt Cardiff, at this moment in time, might be the right club."
Neither player will be eligible for Tuesday's league game between the two clubs.
Solskjaer, the former United striker, took over Cardiff this month.
Manchester United duo Fabio da Silva, Wilfried Zaha headed to Cardiff City
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