That’s according to the Daily Mirror, who claim Clarets boss Sean Dyche has been added to the winger’s growing list of admirers.
Lowe was the Blues’ top scorer last season, scoring 17 goals – promoting both Wigan and Millwall to submit £1.5m bids for the former Hampton & Richmond Borough ace.
Those offers were rejected by Pompey, who value Lowe in the £3m bracket.
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Yet that hasn’t stopped Millwall from pursuing their interest, with The News reporting on Thursday that Kenny Jackett was bracing himself for another bid for his top scorer.
The Mirror claim that Lions boss Neil Harris has since upped that offer to £2.5m.
That remains well short of the Blues’ valuation, with the Fratton Park outfit under no pressure to let one of their prized assets go.
Indeed, Pompey are reluctant to let both Brighton-bound Matt Clarke and Lowe leave this summer – and have the option to extend the latter’s contract, which expires in 2020, for another year.
But if Millwall believe they are in the driving seat if the Blues do decide to sell, then the presence of Burnley may force them to up their game further.