The Ligue 1 outfit, now heavily backed financially by Frank McCourt, are looking for a replacement and upgrade on Bafetimbi Gomis, whose loan spell was not made permanent, and have been linked with moves for Dembele, Giroud and AC Milan forward Carlos Bacca. But Sky Sports report that L'OM are prioritising Celtic's prized asset, who is also believed to be on Chelsea's radar.
Antonio Conte's men are looking to bolster their striking options after missing out on Manchester United-bound Romelu Lukaku, who is primed to complete a deal worth £100m from Everton, while Diego Costa seems to have played his last game for Chelsea.
Chelsea have been heavily linked with Real Madrid's Alvaro Morata, who was expecting to move to Manchester United before news of the Lukaku deal broke, but were considering a £30m move for Dembele during the winter transfer window, according to The Guardian.
Celtic boss Brendan Rodgers could not guarantee that the former Fulham forward will remain at Parkhead beyond this summer, but stressed that none of his stars will leave Glasgow before their Champions League qualifiers.
Everton have also registered an interest in Dembele but are heavily pursuing a deal for Giroud, who jetted off with Arsenal for their tour of Asia on Sunday (9 July). The France international had to make do with a bit-part role at The Emirates Stadium last season, and the arrival of Alexandre Lacazette for a club-record fee will decrease the former Montpellier hitman's chances of game-time with the Gunners next season.
Reports last week suggested that Marseille were the frontrunners to sign Giroud, believed to be valued at around £25m, but their apparent interest in Dembele has dealt Everton a boost in their chase for the 30-year-old who would help replace former Chelsea prodigy Lukaku.
Along with Giroud, Everton are attempting to lure Swansea City's Gylfi Sigurdsson to Goodison Park. The Icelandic international, 27, wants a fresh challenge after helping Paul Clement's men avoid relegation last season and is close to joining Ronald Koeman's side for around £32m.