Trinidad & Tobago international Jones will be out until the New Year with ligament damage suffered in the 3-0 defeat against England.
Beattie, 30, has been a hit at the Blades, scoring 22 goals last season, which helped relaunch a career that had stalled while he was with Everton.
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Top-flight return: Beattie could be back in the Premier League
Keane wants six new players this summer with experience and will try to sign those who know the English game, according to the Daily Mirror.
Beattie, who has five England caps, attracted Middlesbrough's interest in January but boss Gareth Southgate is in the market for midfielders.
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