Mr Cheek set off on a solo attempt to climb the Shimshal Whitehorn (20,000ft) on July 11 but has not been seen since, having meant to have returned on July 15.
Searches by Pakistan's military helicopters last week proved unsuccessful, with the latest search taking place last Thursday, July 31.
Mr Cheek, 28, now lives in Manchester but his family still reside in Gordon having formerly lived near Selkirk.
The Cheek family are now in discussions as to whether to continue searching, with a Foreign Office spokesman saying they are still in regular contact with them.
The research biologist had been part of an expedition - including fellow experienced climbers Bruce Normand and Peter Thompson - that attempted to climb the Distaghil Sar this year, which stands at 25,863 ft.
However, the group was unsuccessful and spilt up afterwards, resulting in Mr Cheek deciding to try to climb Shamshal Whitehorn himself.
Both Mr Normand and Mr Thompson have been involved in the rescue mission, along with fellow climber Don Bowie. But as, yet, they have been unable to locate Mr Cheek, nearly four weeks since he set off to climb Shimshal Whitehorn.
He was advised not to take on the solo climb by his fellow mountaineers because of poor weather conditions but British mountaineering legend Doug Scott said Mr Cheek should not be criticised for his decision.
Mr Scott said: "It's easy for others to say he shouldn't have tried it or gone on his own but it's not their decision - it's down to the climber.
"People take challenges like this and sometimes it doesn't pay off – that can be through overwhelming ambition or just sheer bad luck as it seems to have been in Ben's case. It can be a fantastic feeling, but sometimes it just doesn't pay off.
"That area is avalanche-prone and it can just suddenly catch you. Unless you're very well equipped with a big team and can do snow profiles, there's just no way to predict what the snow will do.
"It will be awful for the family just now. I feel so much for them and my thoughts are with them. I have a very good idea of what they're going through having sat with families of friends who have been in similar situations and it's a painful process, especially not knowing what has happened.
"It's the uncertainty that's so difficult to cope with."
Mr Cheek has previously taken on the Shamshal Whitehorn with Peter Thompson and another climber, Lee Harrison, in 2006. They were forced to abort the expedition after Mr Cheek was injured by falling rocks.
Just this week, 11 climbers were killed when an avalanche hit a group attempting to ascent K2, the second highest mountain in the world at over 28,000 ft, also in northern Pakistan.
Seven climbers survived the avalanche, while four are still missing.