The 27-year-old has come through the ranks with Edinburgh and boasts over 100 appearances for the Pro14 outfit, in that time claiming 11 caps for his country.
McInally recently showed up well for Scotland during the autumn international Test series in the absence of first-choice hookers Ross Ford and Fraser Brown.
Cockerill, who watched McInally also put in a starring performance as he scored in Edinburgh's 50-20 thrashing of London Irish in the European Challenge Cup on Saturday, believes the forward has done enough to force his way into starting contention for the national side.
"He's had a great season, he's worked hard and he deserves to be starting hooker," Cockerill said.
"He captained the side tonight, drove the team well and performed like he was playing for Scotland."