A Conservative MP arrested on suspicion of rape has “voluntarily agreed not to attend the House of Commons for the period of the bail”, Speaker Sir Lindsay Hoyle has said.

In a statement to the Commons, Hoyle also discouraged MPs from using parliamentary privilege to the name the person arrested last month.

Sir Lindsay said: “The house will know that a member has been arrested in connection with an investigation into an allegation of a very serious criminal offence.

“I have received assurances from the member and the government chief whip that the member has voluntarily agreed not to attend the House of Commons for the period of the bail.

“I, the House of Commons Commission and the house service take the safety of our staff and the parliamentary community as a whole very seriously, and ensuring any necessary measures are taken in respect of MPs and employers and staff.”

A spokesperson for the Conservative whips office told Sky News at the time of the arrest: “These are serious allegations and it is right that they are investigated fully. The whip has not been suspended.

“This decision will be reviewed once the police investigation has been concluded.”

Shadow safeguarding minister Jess Phillips criticised the Tories for failing to suspend the whip from the Tory MP in question, a move that would effectively eject him from the party.

Speaking to Times Radio, the Labour MP said: “It is very difficult to comment on the case itself, because it is quite rightly in front of the criminal justice system.

“But what I would say in this matter, is in any organisation – especially one like members of parliament who every day work with vulnerable people, young activists, members of staff – in any other organisation, were this police investigation to be going on, somebody would be suspended while the investigation was taking place.”

Calling the lack of action “shocking”, Phillips said it sent a “terrible message” that senior figures were still able to secure “protection” from their Westminster status.