Tony Kelly, 56, who worked for Ms Spence at Fraser Property Management, told the jury Coats said: "I killed her last Thursday."
He claimed Coats made the remark as they stood in the car park of a cafe in Broomhill, Glasgow.
Mr Kelly said he wasn't sure exactly when this meeting took place but said it was around two weeks after her disappearance.
He was giving evidence for a second day at the trial of Coats, 42, David Parker, 38, Paul Smith, 47, and Philip Wade, 42, who deny abducting, torturing and murdering Ms Spence at a flat at Meadowfoot Road, West Kilbride, Ayrshire, between April 14 and 28, 2011.
Mr Kelly was asked by solicitor-general Lesley Thomson, QC, prosecuting: "What was said by Colin at that meeting?"
He replied: "When we got into the car park he told me he had actually killed Linda the previous Thursday."
Mr Kelly said of his reaction to being told this: "I was kind of numb, stunned. I didn't know what to say or what to ask. It was cut short because Colin's girlfriend and her son started to walk up the car park."
Ms Thomson then asked: "Do you recall what you said before he said that?"
Mr Kelly replied: "It was a discussion of money he might have been due out of the dealings with Juan Carrera. I said the money would be due to her and that's when he told me what he had done."
The solicitor-general asked: "What were his exact words?" Mr Kelly replied: "You don't have to worry about that. I killed her last Thursday."
Mr Kelly was asked if he did anything after being told this and he said: "I went home."
He was asked if he had told anyone. He replied: "Absolutely not."
Ms Thomson asked: "When was the first time you said it to anyone?"
He replied: "It was to the police."
The court heard this was the second time Mr Kelly had met Coats in the cafe following the disappearance of Ms Spence.
Mr Kelly said Coats had become involved after Ms Spence's parents, Jim and Pat, had allegedly been threatened by John Glen, to whom she owed money.
The court was told Coats had been involved in reassuring Ms Spence's parents, who were worried about her disappearance, that they would not be bothered by Mr Glen.
Speaking of the meetings in the cafe, Mr Kelly said: "He wanted to know about Lynda's financial dealings with everyone. I gave him the background with what she had been up to with my clients and the Chinese clients. I didn't know any more about what she had done with John Glen or anyone else."
He added that Coats told him Ms Spence owed him £68,000 and the interest was growing daily.
Mr Kelly said: "At that stage I was shocked with anything I was hearing about Lynda's financial dealings."
He added that Coats seemed to be annoyed Ms Spence appeared to have used money he gave her to pay back Mr Glen in excess of £40,000.
Mr Kelly was asked what Coats had said about Ms Spence and stated: "Just that she appeared to be a liar and a rip-off merchant."
Mr Kelly said he kept quiet about the murder claim as he felt scared.
The witness also told how he was later in the company of Ms Spence's parents, who were going to clear out a caravan their daughter had bought for them.
Ms Thomson asked how he had felt on that trip.
Mr Kelly replied: "Bloody awful. I knew by that stage what had supposed to have happened had happened. I could hardly look at Lynda's mum."
Under cross-examination by Brian McConnachie, QC, representing Smith, Mr Kelly was asked if he could remember a conversation with Coats in the Vanilla Cafe, Broomhill, after Ms Spence went missing.
Mr McConnachie asked: "Did you say something along the lines of 'I'd like to get hold of Lynda and ask what she's playing at'?" Mr Kelly replied that Coats said to him: "You'll need to do that through a medium."
Under cross-examination, Mr Kelly denied he had made up the comment about Coats saying he had killed Ms Spence.
He also denied an allegation by Derek Ogg, QC, representing Coats, that he was "up to his neck in criminality with Ms Spence".
The trial before Lord Pentland continues.