The Scots are in the grip of a front row injury crisis, which has potentially been added to with the news that their most capped player, Ross Ford, suffered a pectoral injury in training last week.
McInally, pictured, has endured a number of injury frustrations himself which have limited the 27-year-old to just nine Scotland caps. The former flanker has long been viewed as a player of great potential but setbacks have stalled his progress.
If Ford is not fit he would be in pole position for the opening Test of a series that includes matches against New Zealand and Australia.
Speaking after Edinburgh’s 37-10 home win over Ospreys on Saturday, Cockerill said: “I think [McInally] should play next week. I think he has been the form hooker in Scotland. He has been really good and has worked hard at his game. On form, he deserves to start for Scotland.
“He is a very good ball carrier, he works very hard in defence, he is a good leader, he has worked hard on his throwing. I just think he has a very complete game. He is a very good set piece player now and his throwing is improving all the time, but he is a very good footballer as well. I think he will suit how Scotland want to play.”
George Turner is the other hooker in the 36-man squad named by Scotland coach Gregor Townsend and could be in line for a place on the bench, a remarkable turn of events for a player who was unsure of where he would be playing at the start of the season.
The 25-year-old was loaned from Edinburgh to Glasgow Warriors and has seized the opportunity offered by a knee injury to Fraser Brown, which has ruled him out of the series.