The Nigerian striker is unsettled on Teesside and has been also linked with a move back to Portsmouth over the summer.
However, Boro are holding out for big money for a player they only signed two years ago for £7.5million and Pompey have cooled their interest and turned their attention to Fredi Kanoute instead.
Eriksson meanwhile, is armed with a £50million warchest and sees Yakubu as the striker capable of improving City's dreadful goals tally from last season.
And according to the Daily Mirror, the 24-year-old has told a 'pal he will "definitely" join City next week'.
Boro have added Tuncay Sanli and Jeremie Aliadiere to his squad over the summer, but Yakubu's departure would leave Gareth Southgate short in the strike department after Mark Viduka also left.
Southgate has made Manchester United's Alan Smith his top target but may move for James Beattie as his club Everton are favourites to land Smith.