The home secretary, Priti Patel, has said she is considering offering police protection for MPs at their constituency surgeries, as a review takes place to “close the gaps” in security in the wake of the killing of David Amess on Friday.

Patel said local police forces had already contacted all MPs to advise them about measures to improve their safety, and a review is taking place involving the House of Commons authorities and the police.

“We need to close any gaps, basically, where we feel that there are concerns,” she told Trevor Phillips on Sky News.

When pressed about how long the review could take, she said: “This isn’t a case of let’s wait for two weeks, three weeks, four weeks; these are immediate protective measures.”

The Speaker of the House of Commons, Lindsay Hoyle, has said he is reviewing MPs’ security “at pace” after the fatal stabbing of the Southend MP on Friday as he held a constituency surgery.

Writing in the Observer, Hoyle said it was crucial to review whether enough was being done to protect MPs, “especially during surgeries”.

Patel listed “practical measures” MPs could already take to reduce the risks they face, including “booking appointments in advance, checking the details of the individuals that you are seeing, checking the locations in advance, making sure that you are not on your own”.

But she conceded tougher measures were also being discussed, including police protection at surgeries, the regular meetings MPs hold to meet their constituents face-to-face.

“There are other options that are being considered, such as when you hold your surgeries, could you have officers or some kind of protection?” she said. She also declined to rule out the use of airport-style security scanners at MPs’ constituency offices.

Security around MPs was tightened in the wake of the murder of Jo Cox five years ago, with many having safety measures such as panic buttons installed at home and in their offices. Constituency surgeries inevitably involve direct contact with voters though, and are often held in community buildings such as churches.

Patel declined to confirm reports that the suspect arrested at the scene, who has been named as Ali Harbi Ali, was previously known to the Prevent counter-extremism programme. But she stressed that an independent review of Prevent was already taking place.

“It is absolutely right through that review programme, that process that’s in place, that we constantly learn, and we improve,” she said.

The shadow foreign secretary, Lisa Nandy, said the response from local police forces in terms of protecting MPs was “patchy”, and some MPs faced far more threats and abuse than others.

“This just keeps happening. And we keep having this debate, and then nothing very much changes,” she told Sky News.

MPs were given a single point of contact at their local police force in the wake of Cox’s murder, to advise them on security. But Nandy said: “There’s a huge disparity between the advice and support that’s offered by different police services around the country.”