Amaechi, who played for the Cleveland Cavaliers last season, has been released by Italian club Kinder Bologna who have paid up his contract for the remainder of the season.
Sheffield have offered pounds 25,000 to Amaechi, which would not take them above the salary cap limit of pounds 170,000 per club.
The 6ft 10in centre has said he wanted to play in England at least for the rest of this season, but was frustrated because the leading clubs were already up to the limit.
Amaechi is one of seven foreign-based players in the England squad which assembled last night for the three European Championship games over the next 10 days, beginning in Denmark on Wednesday.
Last night Amaechi and several other England internationals watched Sheffield upset League-leaders Towers, who had also wanted to sign him.
Amaechi and the Greece-based Steve Bucknall trained with London on Saturday, but by then coach Kevin Cadle had been told by the club management he could not have the Manchester-born centre.
Cadle said: "To get John under the salary cap would have meant releasing one or even two players and that would be too disruptive."Reuse content