The ex-Jags midfielder impressed in a closed- door trial match against Rangers at Murray Park on Tuesday.
Ton managerial team Allan McManus and James Grady will now have another look at the versatile left-sided player in a second closed-door game next week.
The 23-year-old has quit Irish side Dundalk and is free to sign for a new club in January, but is keen to make the Cappielow switch.
McKinlay said: “Obviously I am keen to get things sorted out and hopefully the trial match went well and Morton will have another look at me next week.
“They are very ambitious under Allan and James and I’d love the chance to sign for them. I am a bit short of match practice, but my fitness is good and hopefully I can do myself justice and earn a deal.”
McKinlay is also being tracked by Clyde boss John McCormack, who will have funds available for strengthening in January now that the Bully Wee have the financial backing of director of football Neil Watt. Irish outfit Cork City are also understood to be interested in taking the former Chelsea kid back across the Irish sea.
Morton have also been boosted by the gradual clearing of their injury crisis. Stevie Masterton, Neil MacFarlane and Dominic Shimmin all played in the bounce game against Rangers.
Ton powerhouse midfielder Allan Jenkins is also on the mend after being out for eight weeks with medial ligament damage to a knee.
The 29-year-old, who has not played since the October 17 clash with Dundee, is aiming to be back for the Boxing Day visit of Ayr United.
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