Mrs Cox was shot and stabbed to death in the street outside her constituency advice surgery in Birstall on Thursday.
Labour and Conservative politicians have been advised to talk to local police forces about the security measures they have in place for meetings with the public.
Emails from the whips office of both parties were sent out in the hours after the Labour MP was killed.
Dan Jarvis, Labour MP for Barnsley Central, said: “I know MPs are scared. We’ll be reviewing our security, but I’ll walk through Barnsley today like every Friday.”
His colleague, Jonathan Reynolds, MP for Stalybridge and Hyde, said: “I plan to keep to all my engagements today, including my surgery. I will ensure there is security present however.”
SNP MP Joanna Cherry said she would be holding a two-hour constituency surgery.
Holly Lynch, Labour MP for Halifax, said: “I will never be able to forget what has happened to Jo and, as we’ve all said, it’s incredibly, incredibly sad.
“I think you do have to bear in mind the safety of your staff at all times as well, who are quite often with you when you’re meeting members of the public, and so you cannot avoid taking these risks and concerns very, very seriously but to not let it stop you from being the effective MP that your constituents deserve.”
Labour former minister Yvette Cooper told the programme: “All of us will tell you it’s so important for MPs to be embedded in the community that we represent.
“If you go down to Asda and someone stops you to tell you about a problem with their neighbour or the fact their mum’s back garden is flooded, you write the details down on the back of a till receipt or something like that.
“That kind of daily discussion you have in the constituency is so incredibly important and no-one would ever want to put that at risk.”
Rachel Reeves, Labour MP for Leeds West, said her constituency office was temporarily closed.
She said: “We mustn’t let the actions of this man drive a wedge between MPs and the people we were elected to serve.
“The work of an MP in our surgeries, our work in the community must continue but I think it’s right today that as well as ceasing the campaigning in the referendum that we close our office.”