Police are "satisfied" a body found on Christmas Day is that of Miss Yeates.
They said at this stage they are treating the death as suspicious, but are waiting on the results of a post mortem examination.
News of the grim discovery comes after detectives investigating Miss Yeates's disappearance began looking at similarities with the case of missing chef Miss Lawrence, 35, from York.
It was confirmed that officers from Avon and Somerset had been in contact with North Yorkshire Police about their investigation.
The discovery was made at 9am on Christmas morning by a couple walking their dogs in the Failand area of North Somerset.
An Avon and Somerset police spokesman said: "We are satisfied the body is Joanna Yeates, but this is pending the results of a post mortem which is not expected for at least 24 hours due to the nature of how she was found in such extreme conditions."
Police said the formal identification procedure is yet to be completed, but they have spoken to the 25-year-old's family and explained the situation.
Miss Yeates' father said yesterday he "would be relieved" if the body was his daughter so they would have the opportunity to say goodbye.
David Yeates said: "Of course we don't know for sure that it's her but we assume that it is.
"Nobody else has been reported missing and it's the body of a young girl close to where Jo disappeared. It hasn't been a Christmas for us. I don't know what it's been. It's been surreal, totally unreal. We're just waiting now for the police to tell us what we suspect.
"In truth we feel like we hope ... we would be relieved if it was her.
"We don't want to go to our graves wondering where she is and not being given the chance to say goodbye to her.
"We assume that she's dead so we'd like to say goodbye ... our lives have gone."
The body was found on a snow-covered verge on a country lane on Christmas morning three miles from where Miss Yeates lived with her boyfriend, Greg Reardon.
The scene in Longwood Lane, close to Bristol and Clifton Golf Club, was sealed off for forensic examination.
The Avon Fire Service helped police remove the body from the scene, which was clothed and covered in snow, in order to preserve the site for a thorough forensic examination.
Police are expected to continue their fingertip search in the Longwood Lane area throughout today.
A post mortem examination has been taking place in Bristol today, but due to the extreme freezing conditions in which Miss Yeates body was found, results are not expected until tomorrow at the earliest.
Chief Superintendent Jon Stratford said: "Our heartfelt condolences go out to Joanna's family for their loss.
"We have not stopped working hard throughout the Christmas period to find their daughter after she was reported missing.
"Now we will work just as hard to discover exactly what happened to her and how she came to be in Longwood Lane on Christmas morning.
"Until the post mortem examination is able to firmly establish how Joanna died, we are keeping an open mind about the cause of her death.
"However, I would appeal to anyone with any information whatsoever that can help this investigation to please come forward and help us provide Joanna's parents with the answers they so desperately want and need."
The last known sighting of Miss Yeates, known to her friends as Jo, was on CCTV as she bought a pizza at a branch of Tesco Express in the Clifton area of Bristol on the evening of December 17.
The university graduate, who worked as a landscape architect for BDP, was making her way home from the Ram pub in Park Street at about 8pm after a night out with work colleagues.
On her way home she rang her best friend, Rebecca Scott, to arrange to meet on Christmas Eve.
Her boyfriend, Mr Reardon, 27, reported her missing on Sunday night after returning home to their flat in Canynge Road, Clifton, from a weekend away in Sheffield visiting family.
Friends launched a campaign to find her by putting up posters across Bristol and setting up a website and a page on Facebook and using Twitter.
Officers released footage of her buying the pizza in Tesco, which has become a key part of the investigation.
There was no trace of the pizza, the wrapping or the box in her home - despite the fact that the receipt, the coat she was wearing, and her mobile phone and keys were inside.
Anyone who can help the investigation can call the Operation Braid incident room on 0845 456 7000 or the independent charity Crimestoppers on 0800 555 111.
Tributes have been paid to Miss Yeates on a facebook page set up to help find her.
Bec Wood wrote: "Rest In Peace Jo. You will always be loved and remembered for being so beautiful, kind, successful, and lovely. You made Greg so happy. Our thoughts are with Greg, your parents and all of your families and friends. We will miss you Jo, and hope that you have peace now. Love and prayers xxxxxx"
Michael Whitcher added: "Rest in peace little Jo, we will miss you always. My hearts and thoughts go out to her family, and I am thankful I knew you. God bless xxx".