The World Cup winner suffered knee-ligament damage on January 8 in a European Cup clash against Perpignan and the initial diagnosis put his recovery time at between six and eight weeks.
England, who badly missed Wilkinson's goal-kicking as they slipped to defeat against France on Sunday, travel to Dublin to play championship favourites Ireland on February 27.
Charlie Hodgson and Olly Barkley missed six penalties and a drop-goal between them as France snatched an 18-17 win at Twickenham, leaving both England and Lions supporters pining for Wilkinson's quick return.
But Andrew warned he will not be available for Test duty until the Italy game at Twickenham on March 12.
'Jonny will not be fit in time for England's game against Ireland but he has a chance of being ready for the matches with Italy and Scotland,' he told the Evening Standard.
'Charlie and Olly Barkley are proven kickers in the Premiership but there is a lot more pressure at Test level. We have seen Jonny prove he can handle it in his career but he's not ready yet to return. We are monitoring him closely,' said Andrew.
Wilkinson sustained the injury in Perpignan in only his second match back after severe bruising to his bicep ruled him out of England's autumn internationals against Canada, South Africa and Australia.
Wilkinson also missed the 2004 Six Nations after undergoing surgery to correct a long-standing problem in his shoulder and has not played for England since the World Cup final in November 2003.