McCarthy, who celebrates his 31st birthday today, will continue to be monitored early in the week and a decision will be made on his availability on Thursday.
Flying winger Fetu'u Vainikolo was rested on Monday on his return from international duty with his native Tonga.
The 27-year-old scored two tries during the international period this monring, including the decisive try in Tonga's 21-15 win over Scotland last Saturday.
Meanwhile, Connacht prop Ronan Loughney has not trained this week due to a knee injury picked up during the 14-3 defeat to Dragons last Friday. The results of his scan will determine his involvement this weekend.
Thomas Anderson, Finlay Bealham, Mata Fifita, Eoin Griffin, Frank Murphy, John Muldoon, George Naoupu, David Nolan and Kyle Tonetti all remain on the westerners' injury list and are unavailable for selection.
Tickets for the Connacht v Edinburgh match are available to buy online here. They can also be purchased directly from the Connacht Branch offices in Galway.