The Labour MP and deputy Commons speaker also wants to meet “urgently” with Facebook and Twitter representatives to see if they can do more to protect female MPs online.
The suggestions – which are understood to have support from senior parliamentary figures – come as better safeguards are considered after the brutal killing of Mrs Cox, the Labour MP for Batley and Spen.
However sources said it is unlikely calls to force MPs to take up panic alarms or have extra police protection will be adopted, with some believing a ‘one size fits all approach’ may not prove most effective.
There is growing willingness to take action against the rape threats and violent abuse MPs receive on social media, which has increased since the heated debate on Syrian air strikes last year.
Theresa May, the Home Secretary, willon Monday meet with MPs at lunchtime in Westminster to hear their security concerns after Mrs Cox’s killing.
There will also be a private meeting of parliamentary authorities and the independent body that controls MPs expenses, which offered a new package of security measures in January.