Lucescu brought the 26-year-old to Europe in 2010 when he was in charge at Shakhtar Donetsk, Costa playing over 200 games for the club before moving to Bayern in 2015.
He has continued to impress at the Allianz Arena but has struggled to hold down a regular starting spot under Pep Guardiola and latterly Carlo Ancelotti.
Costa only started 19 games in all competitions in 2016-17 and has previously looked for assurances that more minutes will come his way going forward.
"He's a great player and a good lad, but he needs an environment that stimulates him," the Romanian told Tuttosport.
"If you ask me, I'd tell him to go to Juve.
"That variety forces players to improve and develop. I think Douglas Costa at Juventus would take a definitive leap in quality, but don't let me manage the market.
"He can play anywhere, even as a central 'trequartista'. He's a tactical player, suitable for Italy."