The veteran duo both toured with the Lions in 2005, when a total of eight Tigers made Sir Clive Woodward's original squad.
Murphy, who has been recalled to the extended Ireland party ahead of Saturday's Six Nations clash with England, had attracted interest from provincial sides back home but he has all but agreed to remain at Welford Road.
"I have chatted to the club and I am in the process of ironing it out at the minute," Murphy told Martin Crowson of the Leicester Mercury.
"But I have pretty much agreed to stay. I am just waiting to put pen to paper.
"It is a great club and a club I have known (I would resign) for a long time."
Murphy has been with the East Midlands giants for his entire career, winning a host of English league titles and two European Cups along the way.
And despite having been targeted by other clubs for a number of years, Murphy insists his heart has always belonged to Leicester.
"In this day and age, it is very rare to finish your career in one place," added Murphy, who turns 32 in April.
"I have enjoyed it here and the club has been very loyal to me. In return, I feel loyalty to the club as well.
"It (the new deal) is two years, possibly three. The way I play the game it is important that I keep a little bit of speed, but I feel good at the minute."
England prop White, who featured in all three Tests for the Lions against the All Blacks five years ago, has signed an extension that will see him continue playing until at least the end of next season.
And with White's birthday falling in May, his new deal will mean he will be nearing 38 before he entertains any thoughts of retirement.