It is the second time the 43-year-old has been viewed as a potential Chelsea manager by Abramovich - in 2004 he eventually lost out to Jose Mourinho whose Porto side beat Monaco, coached by Deschamps, in the Champions League final.
However, according to the Telegraph Abramovich - who sacked his sixth manager since he bought the club in 2003 when Andre Villas-Boas was shown the door in early March - apparently held discussions with Deschamps a fortnight ago in Paris.
It is thought that Deschamps has re-entered the equation because Abramovich believes it is unwise to bring back Jose Mourinho from Real Madrid, as it would suggest weakness on his part after sacking him in 2007, and he believes it unlikely that Barcelona's Pep Guardiola will agree to move.
Abramovich is understood not to favour giving the full-time job to interim manager Roberto Di Matteo, even though the Swiss-born former Italian international has rescued the club's season in taking them to both the FA Cup and Champions League semi-finals.
Deschamps, who was sounded out by Liverpool two years ago, has impressed both at Monaco then in returning Juventus back to Serie A and latterly at Marseille whom he guided to the 2010 league title.
However, Deschamps - who played for Chelsea in the latter days of his career winning the FA Cup with them - is said to be increasingly disenchanted at Marseille.
Unlike former Chelsea boss Carlo Ancelotti at PSG he has had to battle for transfer funds and despite a League Cup final to look forward to they look unlikely to qualify for next season's Champions League.
Morale is at an all-time low as Marseille are on a club record run of 10 matches without a win leaving them in ninth spot in Ligue 1 some 16 points off Lille, who occupy the third and final Champions League spot.
Source: AFP
Source: AFP