Solskjaer was seen as a stop-gap option when recruited as a caretaker in December, with United scrambling for a solution following the sacking of Jose Mourinho.
Solskjaer, who scored 126 goals in 366 appearances for United, was already a hero to fans as the man who scored the stoppage-time winner in the 1999 Champions League final to complete the treble.
Solskjaer has made a remarkable impact since arriving as United's interim boss in December following Mourinho's sacking.
Mourinho has been out of work since being sacked by Manchester United in December 2018.
The CFA made no mention of the 45-year-old Cannavaro, who has been strongly linked in Chinese media with the post.
He was speaking at a press conference ahead of Tuesday's meeting with Ajax at the Bernabeu in the Champions League, a competition Real have won the last three years.
Mourinho has been among those mentioned as a possible replacement for Solari, as Madrid have been knocked out of both the Copa del Rey and, realistically, La Liga's title race within four days.
The amount, listed under "exceptional items" in the club's second-quarter results, includes pay-outs for the members of Mourinho's staff.
A recent report in Britain's Sunday Telegraph said Southgate was on a shortlist being drawn up by Old Trafford chiefs as they sought a permanent replacement for the sacked Jose Mourinho.
Mourinho, sacked by Manchester United in December, told BeIN Sports how he skirted a touchline suspension on the eve of the 2005 quarter-final against Bayern Munich
Substitute Romelu Lukaku broke the deadlock just 38 seconds after coming on midway through the second half.
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Solskjaer, currently in charge of Norway's Molde, was widely reported to be United's leading candidate for the interim role for the rest of the season.
Mourinho was dismissed on Tuesday after United's worst start in 28 years left them trailing Premier League leaders Liverpool by 19 points.
Mourinho was dismissed by United on Tuesday after a poor run of form left the club sixth in the Premier League table and 19 points adrift of leaders Liverpool.