The move comes less than 24 hours after the troubled Gorgie club announced assistant boss Billy Brown would not be offered a new deal when his short-term contract runs out at the end of the month.
Hearts entered administration with debts of around £30million last summer but fans' group the Foundation of Hearts is hopeful of completing a rescue deal in the next few weeks.
However, there are still cash-flow concerns and Southern and Murray have now followed the lead of first-team players such as Ryan Stevenson, Jamie MacDonald, Jamie Hamill, Kevin McHattie and skipper Danny Wilson by accepting reduced payments.
Bryan Jackson, the joint administrator of Hearts and business restructuring partner with BDO, said: "Both David and John have kindly accepted a cut in wages to help reduce the running costs at the club.
"We remain focused on the main issue of getting Hearts out of administration by negotiating with the Lithuanian administrators for the transfer of shares.
"However, a key element in the success of this administration is keeping Hearts solvent throughout this process until the share transfer can occur and all issues can be resolved.
"I would like to thank the team and staff at Hearts and the fans for their patience and loyalty throughout a very difficult period. We are getting closer to a resolution but there is still some way to go and several obstacles to overcome so I would ask everyone to remain focused on the end result which is to keep the club going both on and off the pitch for the next few months."
Hearts confirmed on Tuesday night that assistant boss Brown would be leaving the club.
The 63-year-old - a former number two to ex-manager Jim Jefferies - returned to Tynecastle during the summer as he volunteered to help new boss Gary Locke through pre-season by working for free.
He then signed a short-term deal until January 31 but BDO have now decided they can not afford to keep Brown on past that date.
A Hearts spokesperson said: "We would like to put on record our gratitude to Billy for his sterling efforts in helping Gary since his arrival in July.
"He has been of considerable assistance to Hearts over the last six months and we know he appreciates the need to refresh the team coupled with the need for continued financial prudence. We wish him well in the future."
Brown will take his place in the dugout for the final time for Saturday's Premiership meeting with St Johnstone at McDiarmid Park, officially leaving the club on Sunday, January 19.
Hearts have confirmed they will not be seeking a replacement.
However, Brown urged the Jambos supporters to continue backing Locke after the league's bottom side dropped 20 points behind nearest rivals Ross County, saying: "I am very disappointed my contract will not be renewed but that is the way things are at the club and I have to accept that.
"I was really pleased when I was asked to assist Gary at the start of the season.
"Having worked with Gary as a player over a great number of years I was always of the opinion that he had the making of a top manager and I've seen nothing to change my mind.
"The way he has handled a very difficult situation has been magnificent.
"I urge all supporters to get behind Gary and the team for the remainder of the season and would like to thank everybody who has helped me in my third spell at Hearts, especially the players and staff whose spirit remains remarkable."