The Dutch forward sustained a hip injury during the Premier League match with QPR last time out but, as exclusively revealed by Goal.com, the Red Devils were confident the ex-Arsenal man would recover in time for the last-16 second-leg.
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The Scot told reporters: “Robin van Persie has trained this week – he’s available for tomorrow [Norwich]. His hip was very tender but he’s okay now."
The United boss was less optimistic about Jones' chances of making the crucial European fixture, suggesting he is unlikely to make the game, but did rule him out of the match with the Canaries at Old Trafford.
“Phil Jones starts running today," he continued. "He’s got a fight on his hands to be ready for Tuesday [Real Madrid] but we’ll give him every chance. He won’t be available for tomorrow [Norwich]."
The current Premier League leaders go into the game with Chris Hughton's men a massive 12 points ahead of Manchester City in the title race.
Looking ahead to the match, Sir Alex warned his players not to get complacent and insisted that they cannot afford to be distracted by the "occasion" surrounding the return leg with Madrid's.
“Tomorrow is an important one because Tuesday will look after itself," he added. "It’s an occasion on Tuesday whereas tomorrow is the mundane practical situation in that we need to win.”
Meanwhile Ryan Giggs has been rewarded for his continual good form with a new one-year contract at the club, a deal that will see him play in his 23rd senior season at Old Trafford.
Sir Alex hailed the attacker's achievement's and indicated he will hand him his 1000th career appearance, including matches for Wales and Great Britain, in the very near future.
He said: “He’ll be involved [against Norwich] and will be involved on Tuesday [against Real Madrid]. I don’t think what he has achieved will ever be achieved by anyone again."