Mrs Cox, 41, was shot and stabbed outside her constituency surgery in West Yorkshire on Thursday. A 52-year-old man has been arrested.
It is understood that a Nazi regalia was recovered at suspect Tommy Mair’s home.
David Cameron and Jeremy Corbyn have described the killing as an “attack on democracy”.
West Yorkshire Police’s temporary Chief Constable Dee Collins said the force was also pursuing the suspects link to mental health services.
She said the man arrested, named locally as Mr Mair, had been examined by two specialist medical practitioners and deemed fit for detention and interview and said detectives were continuing to question him.
The visit by the Prime Minster and Labour leader came as the Conservatives, Lib Dems and UKIP all announced they would not contest the by-election resulting from her death.
Joined by Commons Chaplain Rose Hudson-Wilkin, Leeds Central MP Hilary Benn and Commons Speaker John Bercow, the prime minister and the Labour leader bowed their heads as they laid bouquets in Birstall.
Mr Corbyn said he had asked for Parliament to be recalled to enable politicians to pay tribute to the Labour MP “on behalf of everybody in this country who values democracy… free from the kind of brutality that Jo suffered.”
He added: “Jo was an exceptional, wonderful, very talented woman, taken from us in her early 40s when she had so much to give and so much of her life ahead of her.
“It’s a tragedy beyond tragedy what happened yesterday.
“In her memory, we will not allow those people that spread hatred and poison to divide our society, we will strengthen our democracy, strengthen our free speech.”
Vote Leave and Remain have both suspended campaigning in the EU referendum in light of the attack.
Mr Cameron said: “Where we see hatred, where we find division, where we see intolerance we must drive it out of our politics and out of our public life and out of our communities.
“If we truly want to honour Jo, then what we should do is recognise that her values – service, community, tolerance – the values she lived by and worked by, those are the values that we need to redouble in our national life in the months and years to come.”
Commons Speaker John Bercow described her as an “outstanding” MP.
“I think everybody is united in grief, in horror and in an overpowering respect for somebody who we came to know. Whose talents we admired, whose passion we observed on a daily basis,” he said.
The leader of the Liberal Democrats, Tim Farron, said she was “inspiring to listen to”, adding: “She was an outstanding representative who stood up for her community diligently.
“A young family has lost a mum and a wife, a community has lost a great MP and Parliament has lost an authentic and passionate voice.”
The Batley and Spen constituency Labour Party said Mrs Cox had “meant the world” to them.
The party added: “She was a lovely woman who cared deeply about her family, her community and everyone around her.
“She was intelligent, but so down to earth, and anyone who met her for the first time felt like they had known her for years.
“Above all, she was an amazing wife to Brendan and mum to her two children. It is a tragedy that they will have to grow up without their lovely mum.”
Witness Ben Abdullah, who was working at a café next to the scene of the attack, said he saw “a river of people” coming down the street “screaming and shouting”.
Mr Abdullah said he heard several shots and saw Mrs Cox on the floor “in a very bad state”.
Politicians have been warned to review their security in the wake of the attack
On Thursday, hundreds of people of all faiths packed into Saint Peter’s Church in Birstall for a service of remembrance while a vigil was also held outside Parliament.
Mrs Cox is the first sitting MP to be killed since 1990, when Ian Gow was the last in a string of politicians to die at the hands of Northern Irish terror groups.
She was married to campaigner Brendan Cox, and had two young children, with the family dividing its time between its constituency home and a river boat on the Thames.
Mr Cox said in a statement: “Jo would have no regrets about her life, she lived every day of it to the full.
“Jo believed in a better world and she fought for it every day of her life with an energy and a zest for life that would exhaust most people.”
The family and friends of Mrs Cox have set up a fund to raise money for three causes which they said were close to her heart. Within hours the fund had exceeded £100,000.
Mr Cox tweeted: “Protecting people in Syria, tackling loneliness & fighting extremism; some of the causes Jo fought 4.”