Lennon hopes the deal will be concluded shortly and hopes the 27-year-old could even join up with the rest of the squad on Celtic's tour of North America.
The Hoops boss was speaking after their 1-0 defeat to Philadelphia Union, as they prepare to take on Manchester United in Toronto.
The capture of Juarez from UNAM Pumas also looks imminent.
"Juarez is pretty close, " Lennon told philly.com
"We think we're pretty close with Murphy from Sunderland as well. I think so."
Asked if Murphy would be joining Celtic in America, Lennon replied: "If we can get it done, yeah."
He added: "He's quick, he's six foot two and I've watched him a lot for a few years now.
"He was playing for Ireland in an end of season tour in America, he caught my eye and I've kept an eye on him ever since.
"I think he'd do all right, he could take the hits for us up here."
Murphy joined Sunderland in 2005 and had a loan spell with Ipswich last year, scoring six times in 18 appearances.
While Lennon was preparing to welcome new faces to his squad ahead of the new season, the club paid tribute to Artur Boruc, following his move to Fiorentina.
The 30-year-old goalkeeper has quit Parkhead after five years to pen a two-year deal with the Italian club.
Celtic chief executive Peter Lawwell told the club's official website: "It was clear that, after five years with Celtic, Artur wished to pursue a fresh challenge.
"I know everyone at the club and all our supporters will sincerely wish Artur all the very best in this challenge.
"During his time at Celtic, he has been a magnificent servant to the club and, without any question, he is one of the finest goalkeepers ever to play in Scotland.
"Indeed, he has been one of Europe's top keepers.
"A true character, a passionate man and a real Celtic supporter, he has been part of some memorable occasions and been instrumental in the successes which the club has enjoyed.
"We send him all our best wishes for the future."