McEleney, who plays as an attacking midfielder for League of Ireland side Dundalk, is out of contract at the end of December.
And though Blackburn, Coventry and Doncaster have all been linked with the 25-year-old's services, Oldham have moved into pole position after offering him a contract, according to the Irish Sun.
The report claims that McEleney is in line to replace Jack Byrne - a former Blackburn loanee - at Oldham, with the Wigan player's temporary deal also up at the turn of the year.
McEleney though is also said to have offers on the table from Derry City and Shamrock Rovers, and previously cut short a spell at Sunderland in his youth, after suffering from homesickness, according to the Irish Sun.
The report claims Ipswich Town previously took a look at McEleney, who has scored 10 goals in 38 games in 2017, but opted against bringing him to Portman Road.
Would McEleney be a good signing for Oldham/Blackburn/Coventry/Doncaster?