Sunderland have completed the signing of Celtic right-back Adam Matthews for a fee of £2million.
Diego Poyet is determined to establish himself in Slaven Bilic's West Ham side this season and avoid another loan spell.
The pair have long been linked with a move to Stade Louis II, with the ambitious club looking on course for promotion back to Ligue 1.
Monaco are backed by Russian billionaire owner Dmitry Rybolovlev and reports have suggested moves for Beckham and Lampard are in the pipeline.
Ranieri signed the latter for Chelsea during his time at the Stamford Bridge helm and asked if would like to sign Frank Lampard, he said with a laugh: "Oh, it would be fantastic.
"[He could perform well in Ligue 1] - why not? Frank could play in every country."
Ranieri, speaking to ESPN STAR Sports, admitted he was unsure as to whether an offer has been made for Beckham.
The 37-year-old recently ended his spell at Los Angeles Galaxy and is considering his next move.
"Oh he is the same - he is a big champion," Ranieri said. "He would be a good professional player for us.
"I like all the champions and he is a champion."
Ranieri did, though, suggest a move for both veteran midfielders may be too much for the club's owners.
"Look, I think that my chairman wants young people," he added. "Young players to develop that can get better [together over time]."