The two soldiers were on a training exercise on the hottest day of the year in Powys on Saturday.
A third is in a serious condition after Saturday's incident which is thought to involve six soldiers collapsing in hot weather.
The Ministry of Defence (MoD) refused to comment on reports.
However, the BBC understands the two reservists were taking part in the selection process for the Territorial SAS.
They would have already carried out training over a period of weeks and were believed to have reached the early stages of the final selection stage.
It would have been a supervised exercise, but it is likely it would have involved them acting on their own moving from checkpoint to checkpoint.
The Brecon Beacons is home to the Infantry Battle School and makes up one of Britain's largest military training areas.
An investigation is focusing on the weather conditions and the nature of the exercise.
The deaths occurred on the hottest day of the year so far in Wales, with temperatures reaching 30C (86F) in Powys.
Phil Speck, who was walking in the Beacons on Saturday, said he saw some soldiers struggling in the heat.
“We saw many soldiers across our walk – probably a dozen or so,” he said.
“To start with they looked fine. Towards the end of the day they were feeling the heat. One soldier spent quite a period of time recovering.”
The soldiers appeared to be in full kit and he said some “were clearly tired and exhausted”.
“When you see a soldier you kind of assume that they are fit enough to deal with the situation,” he added.
Mr Speck, from Abercynon, Rhondda Cynon Taf, said he is an experienced walker and he and his friends decided to avoid some peaks because of the heat.
It is understood live ammunition was not involved in Saturday's exercise.
The MoD said it was working with Dyfed-Powys Police.
A statement added: “The two servicemen's next of kin have been informed. More information will be released in due course but it would be inappropriate to comment further at this stage.”
The Brecon Beacons national park is used for military training because of its relative remoteness.
The infantry regiments of the British Army train at Sennybridge in the area and there is an Army base in Brecon.
Conservative MP Colonel Bob Stewart – a former British Army officer and member of the Defence Select Committee – said the soldiers would have been largely unsupervised and the symptoms would have come on very quickly.
He added: “They were probably going cross country, up and down the mountains on their own, which makes it even more difficult to identify heat exhaustion or heat stroke because, of course, they've got no friends around them to do something when they collapse.”
Kirsty Williams, leader of the Welsh Liberal Democrats and AM for Brecon and Radnorshire, said it was a “meltingly hot” day in the area on Saturday.
She added: “It's very, very sad to think that people that were serving with our armed forces have lost their lives in this way.”
Ms Williams said it was a beautiful area which could also be very “treacherous” and the military trained there “for a reason”.
She said she hoped “any lessons that can be learned from this tragedy will be learned”.
About 30 members of the four south Wales mountain rescue teams (MRTs) were called out to help in the emergency on Saturday near Pen y Fan.
Col Mike Dewar, a defence analyst, told BBC Radio Wales he found it difficult to understand what happened in apparently “perfect conditions”.
He said the training session may have been pre-booked a long time before, and the soldiers would have been warned to take plenty of water in the hot weather conditions.
He said “all sorts of training takes place there” – from special forces to regular servicemen and women – and the soldiers may have been carrying heavy packs.
Col Richard Kemp, former infantry commander, told the BBC Breakfast programme that the Brecon Beacons were valued by the military for training because they were hard to navigate and cross on foot and provided extremes of heat and cold.
He said soldiers on special forces selection were likely to be operating alone which made it harder for signs of heat exhaustion to be spotted quickly.
He said: “When you begin to get engulfed by heat illness, your thought processes blur.
“The staff at the check points should be looking to see if anybody is having particular problems.”
Mark Moran, from Central Beacons MRT, said: “We were working alongside military personnel who remained extremely calm and professional during this tragic incident.
“Our thoughts are now with the families of those involved.”
Mayor of Brecon and Powys county councillor Matthew Dorrance said: “It's incredibly sad for the friends and family of the people who have lost their lives and thoughts are with the person who is injured.
“In one way we've been blessed with the weather but for people working in this heat, they're tough conditions.”
Mr Dorrance said local people regularly saw troops training in the area and were “proud of our links with the military”.
The MoD said it was working with Dyfed Powys Police to investigate the incident.
“The two servicemen's next of kin have been informed. More information will be released in due course but it would be inappropriate to comment further at this stage,” said a spokeswoman.