The 21-year-old has netted three times in 11 league outings for the Canaries since joining Norwich at the end of November, helping City maintain their push for promotion to the Premier League with Lambert's men currently fourth in the Championship standings.
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Lansbury enjoyed an equally profitable spell on loan at Watford last season, scoring five goals in 37 league appearances.
As a result, the starlet has featured for the Gunners' first team once this season, with Lansbury starting, scoring and getting booked in the 4-1 victory over fierce rivals Tottenham Hotspur in the League Cup.
With Emirates Stadium supremo Arsene Wenger seeing enough potential in Lansbury to throw him into senior action alongside Samir Nasri, Jack Wilshere Denilson and Tomas Rosicky, Lambert concedes that the player is more likely to be set for stardom with the Gunners, rather than a permanent deal with Norwich.
"I think Arsenal are the main players in that one," he told Yahoo. "He's a Barclays Premier League player.
"He's at a terrific football club, and he's learning all the time. I think we're a long way from being able to ask Arsenal if we can take him permanently. He might have a big future at Arsenal."
Paul Lambert was speaking to Yahoo! through its partnership with the League Managers Association. Visit The Dugout at www.yahoo.co.uk/dugout
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