Gilmour's earlier stand-out display against England has raised the number of Premier League clubs - including newly promoted Norwich City - who are keen to take him on loan next season.
While that might be the next stage in the gifted youngster's development, Kovacic is convinced Gilmour has everything it takes to then become established at Chelsea.
"Unfortunately I didn't get the chance to play against Billy but I did talk with him on the phone as soon as it came out that he was positive," said Croatia midfielder Kovacic.
"Of course I was very sad about that because he is doing excellently and he is my team-mate. I felt sad for him.
"He has had an excellent season and his qualities cannot even be brought into question. He is still a young player but he has played a lot of good matches for Chelsea and for his national team.
"I don't give him too much advice, but we do talk a lot,' added the 27-year-old. 'I mean, we are together every day at our club.
"I have seen Billy's abilities from the first time he started training with the first team. He can become a really excellent player, so it is up to him to work hard and to get better with each day. I am sure that, with his personality, he is going to do that."
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