The 19-year-old midfielder, who is a boyhood Villa supporter, is set to become a free agent next summer, alerting the Blues to the young talent.
Villa teenager Grealish has made three appearances in all competitions for the Birmingham side this season, continuing to impress at Villa Park.
And manager Lambert feels Grealish, who has been capped by the Republic of Ireland at Under-21 level, would be best-served staying at Villa.
“This is the best place to be because he’s getting the chance to get round and about our first team,” said Lambert.
“He’s playing little bits here and there but he’s still far from being an establish player. He’s not near the finished article. But he’s developing, which is great. The way his career is going hopefully something will be done.”
Grealish is yet to decide his international future despite racking up six appearances for Ireland’s Under-21 side.
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