Cole, 35, left Roma on Tuesday after his contract was terminated by mutual consent, having failed to feature for the Serie A side since last March following a difficult first eight months at the Stadio Olimpico.
Reports have subsequently claimed that he has already agreed terms with the Galaxy [h/t BBC], but while Klein acknowledged that the club are in advanced talks with the former Arsenal and Chelsea defender, he was also quick to stress that nothing has been finalised as of this moment.
“Still very close. Nothing new to report,” he told MLSSoccer.com. “There's other issues to work through, but he's a player that has his own qualities, and we're getting closer. But it's not done yet.”
Cole is certainly not the only European player currently in talks with the Galaxy either, as Belgian defender Jelle van Damme also looks set to arrive at the StubHub Center in the near future.
Standard Liege manager Yannick Ferrera revealed earlier this week that his club captain was on the verge of joining the MLS club, but much like the Cole deal, Klein stressed that there are still a few details to be ironed out before the move can be made official.
There may even be a third high-profile signing if everything goes to plan, with the Galaxy chief admitting that there is “some truth” to reports linking them with a move for AC Milan midfielder Nigel de Jong [h/t LA Times].
The 31-year-old Dutchman has only played six times for Milan so far this season due to injury issues, and despite signing a contract extension until 2018 last June, is now believed to have been deemed surplus to requirements.
LA are looking for a defensive-minded midfielder to partner Steven Gerrard in the centre of the park after selling Juninho to Mexican side Club Tijuana last month for around $1.5 million [£1.02 million], and Klein told MLSSoccer.com that the 81-times capped Netherlands international is a potential candidate to fill the role.
“I would say where there's smoke, there's fire, so there's some truth,” he said when asked about Van Damme and de Jong. “Two different stages [of negotiations]. I would probably put Van Damme in the Ashley Cole camp. And de Jong is, obviously, a fantastic player and kind of a wish-list type of guy, and he fills a positional need for us, and we'll see what happens.”