The future Chelsea midfielder produced a Man of the Match performance for Scotland in their recent draw with England at Euro 2020.
The Wembley stalemate gave the Scots real hope of progressing to the knockout stages of the European Championships for the first time but they were then beaten by Croatia in a game Gilmour missed after testing positive for coronavirus.
But while the Euros ended badly for Gilmour, Kovacic believes the future is bright for Gilmour, tipping him to establish himself as an ‘important’ player at Chelsea over the next few years.
Kovacic said: ‘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.
‘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.
‘He is definitely going to be one of the better players for Chelsea and an important player for Chelsea in the near future.’
He added: ‘I don’t give him too much advice, but we do talk a lot. 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.’
However, in order to guarantee Gilmour more playing time, Chelsea are considering loaning him to newly-promoted Premier League club Norwich City for the 2021-22 campaign.