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The new manager and acting chairman have joined forces to try and convince the player, who was appointed club captain by former boss Steve Claridge, to see out the remainder of his contract.
Livermore handed in his transfer request almost immediately after Claridge's sacking last week, citing his disillusionment with the club and the way it was going as his reasons for wanting to leave. But having already convinced goalkeeper Andy Marshall to stay, Paphitis is hopefuly of doing the same with Livermore.
Speaking to the South London Press, Paphitis said: "We're doing our best to keep David.
"We're speaking with him and his agent, that's myself and Colin Lee. Obviously we can't force a player to stay, but hopefully he'll see that under Colin Lee we'll be heading in the right direction as a club.
"Andy Marshall had told us he wanted to leave before, but he now says he's been impressed after working with Colin Lee in training and is happy to stay for a year. And we'll review his contract situation at the end of the year. "We want the same sort of response with David Livermore too."
Paphitis was also quick to praise manager Lee afor turning things around so quickly.
"Colin's done a fantastic job in stabilising the club already in the short space of time he's been here," said Paphitis.
"We're looking forward to the start of the season now."
Lee, meanwhile, is on the lookout for new talent after admitting that The Lions' rich well of youngsters appears to have dried up.
"There is talent at the club - but there is not enough of it," Lee said.
"Millwall have a fantastic record of bringing youngsters through, but we are not blessed with that as much as we have been.
"The younger players are not as talented as what we've been used to at this club."
Lee is also hopeful of bringing in one or two new faces before the new season starts this weekend after missing out on John Oster yesterday.
"I'm still pursuing targets and hopefully I'll have one or maybe two in before the Leeds game on Sunday.
"I've been given the green light to get three in but obviously there's a limit to the cost of that. Maybe the short-term option may suit us, where I get a player in on a contract up until the next transfer window. "