Earlier, Sheikh Ali Qarsha, a mediator between the Yemeni authorities and the Houthis, also confirmed to Al Jazeera that the Houthis were committed to handing over the Saudi soldiers.
Fighting between the Houthi group and Saudi forces has raged since the Yemeni fighters moved into Saudi territory in November.
The six-point ceasefire agreement also calls for Houthi fighters to unblock roads and return territory to government control.
The government has moved to extend state control into Houthi-held areas.
Al-Salami said that Houthi representatives in the committee had helped organise efforts to clear mines and unblock roads, making it possible for the Yemeni army to deploy to the Saudi border region.
The Houthi fighters, whose main battle has been with the Yemeni government, agreed on Thursday to a truce to end a conflict that has raged on and off since 2004, with fighters complaining of social, religious and economic discrimination.
The ceasefire has proved shaky, with the car of Mohammed al-Qawsi, Yemen's interior ministry undersecretary, came under fire on Friday in the northern city of Saada, hours after the ceasefire was to have officially begun.
The ceasefire itself comes after Western powers ramped up pressure on Yemen to deal with its internal conflicts, which include the Houthis and a secessionist movement in the country's south.
Riyadh and Western powers fear Yemen may become a failed state and that al-Qaeda could exploit the chaos to use the country as a base for attacks in the region and beyond.