Officers are preparing to bring in another team of dog handlers at the weekend to try to find Harjinder 'Harry' Takhar.
Harry was last seen in the Stirchley Pools area, off Stirchley Lane, at 8.30am on Sunday. Search and rescue teams have been in the area since Thursday.
Meanwhile, it emerged that Mr Takhar was with his family when he went missing and climbed over his son in the back seat to get out of the car before bolting off into the woods.
Detective Inspector Jo Whitehead from Malinsgate Police Station, who is leading the investigation, said they were extremely concerned about his whereabouts and have had a full team of detectives working on the case since the 58-year-old went missing.
She said: "Unfortunately every lead we have had so far hasn't come to any fruition. I would appeal for anyone who might have seen Harry - who was well known in the area - to get in touch as soon as they can.
"This has been a very unusual missing person case, as usually we would have had some sort of information or positive sightings since he was last seen. There are a lot of dog walkers in the area who have told us they are familiar with him but unfortunately there have been no sightings.
"We will continue to search the surrounding area and will if necessary be bringing in another dog team over the weekend but we hope he reappears before then."
Harry's has lived with his wife Ran, son Callum and two other children in Dawley for three years. Callum and one of his daughters were in the car with him on Sunday morning when they said he felt sick and claustrophobic and that he couldn't breathe.
Ran said: "It was like a panic attack whilst the car was moving, and when I did manage to pull up he climbed over Callum to get out of the car instead of going through the door on his side, hurting his arm in the process.
"We thought he was going to be sick and then get back in but there is a hedge where you can get access to the field and he just bolted. We were in shock but Callum and my daughter eventually chased after him.
"He walks the dogs regularly in this area and probably knows every inch of the park. I feel and know in my gut that he is not in there and that he left on the Sunday morning."
Mrs Takhar added: "Harry hadn't been sleeping too well and was quite anxious. He was on new medication which heightened that, so I feel it may have been a factor in him having a serious panic attack and feeling he had to flee.
"This is totally out of character for him he has never done anything like this before so something has gone seriously wrong.
"We just need a positive sighting that he has actually left the park, whether that be on CCTV or whether it be from somebody getting in touch with the police, who have been wonderful." "Everyone knows Harry at the park because of the dogs, even though we have only been here three years. We feel someone must have seen him at some point, although he seemed to leave very quickly."
Meanwhile, a campaign to find any information on Harry has been launched in the West Midlands where the family have links in the Smethwick, Bearwood and Greater Birmingham areas,
Mrs Takhar said: "Because he was ill I think he may have fled to his old home and everyone has been fantastic around there, with over 3,000 leaflets distributed in taxis, shops and temples, the word has really spread. A Facebook campaign has been launched as well."
Detective Inspector Whitehead said that when Harry was last seen he was wearing a navy-blue Barbour jacket, navy blue shirt, black jeans and black trainers.
She asked for anyone who may have seen him or has any information to call 999 immediately.