Data from a few days ago suggested London was two weeks behind those areas, but now it is said to show the gap has closed to two or three days.
Mr Khan said he would meet council leaders on Monday and then recommend counter measures to ministers.
Health Secretary Matt Hancock has said he would not rule out new restrictions in London to slow the virus's spread.
"The situation is clearly worsening," a spokesperson for Mr Khan said.
"Sadiq will meet council leaders tomorrow and any London-specific measures will be recommended to ministers following that.
"The mayor wants fast action as we cannot risk a delay, as happened in March.
"It is better for both health and business to move too early than too late."
Measures in place in north-east and north-west England include a ban on meeting with other households or support bubbles in private homes and gardens, not socialising with those outside your household in public venues and using public transport only for essential purposes.
Pubs, restaurants and cafes are table service only and all leisure venues must shut at 22:00 BST. Care homes are closed to non-essential visitors except in end-of-life circumstances.
Mr Khan is also said to be looking at the possibility of asking those who are able to work from home to do so, which is in opposition to the government's policy of urging people to get back into offices.
A mayoral source told the Press Association: "It's clear that cases in London are only moving in one direction, we are now just days behind hotspots in the North West and North East. We can't afford more delay.
"Introducing new measures now will help slow the spread of the virus and potentially prevent the need for a fuller lockdown like we saw in March, which could seriously damage the economy once again."
Mr Hancock, who said he had spoken with the mayor over the weekend, told Times Radio he would not rule out advising people to work from home from some point next week.
He told the BBC's Andrew Marr show that
Covid-19 restrictions in England would get tougher if rules were not followed.
