The Croatia international has not been used by Liverpool this season and was left out of the squad entirely for the Premier League opener against Norwich City and the UEFA Super Cup clash versus Chelsea.
Roma was considered the favourite to sign Lovren, only for a deal to reportedly collapse as the sides were unable to agree a fee.
However, Lucci suggested there is still hope of a move going through before the Serie A transfer window closes.
"Lovren is very open to joining Roma and I think that's obvious, seeing as we are talking about a great club," he said.
"After Gianluca Petrachi [sporting director] and Guido Fienga [CEO] took over, you can tell there is a new and far healthier atmosphere at Roma, along with long-term planning for a great future.
"Every operation has its problems, especially for a player of Lovren's calibre who reached the Champions League final in the last two seasons.
"At times, you can overcome these obstacles. I don't know if Lovren will join the Giallorossi, but as things stand, that would certainly be his priority.
Lovren's recent absence has been put down to illness and he spent time on the sidelines last season with pelvic and hamstring injuries, but Lucci stressed the 30-year-old is ready to play.
"I want to counter any stories of fitness issues," he said. "At the start of last year, after an entire season and reaching the World Cup final with Croatia, he had pelvic strain issues that kept him out for the initial part of the campaign, but that's all."