The Spain international moved to Old Trafford from Chelsea for a club-record fee of £37.1m during the January transfer window.
Despite the midfielder having only played in two games for the Red Devils so far, Giggs is confident that the 25-year-old can become a legend at the club.
When asked by MUTV if Mata could be regarded as an all-time United great in the future, Giggs said: "I think he could be. He's obviously got the talent.
"I don't want to put too much pressure on him, but I don't see why not. The players are just like the fans.
"When they see a quality player come through the door, it gives them that excitement and you feel it around training. It gives everybody a lift."
United are currently seventh in the Premier League table, 15 points adrift of leaders Arsenal.