Nineteen young people have met violent deaths in London this year.
Ministers will take "any legislative measures" necessary to tackle knife crime, Gordon Brown has insisted.
Speaking on the final day of the G8 summit in Japan, the prime minister said it was "completely unacceptable" for young people to carry knives.
"We will consider any new legislation that is necessary to get knives off our streets," he told reporters.
Tory leader David Cameron wants anyone caught carrying a knife without good excuse to expect to go to prison.
There have been a number of high-profile knife murders in London recently; 19 young people have met a violent death so far this year.
The latest victim, 14-year-old David Idowu, died on Monday three weeks after being stabbed in south-east London.
Mr Brown said: "We will take any legislative measures that are necessary to deal with knife crime in our country.
"We want to make it completely unacceptable for any teenager in particular to carry a knife. We want a presumption to prosecute anybody carrying a knife.
"We have of course had further legislation that would ban knife sales to people under 18. We are prepared to enforce the legislation but we will consider any new legislation that is necessary to get knives off our streets."
Mr Brown has argued that anyone over 16 caught with an illegal knife should be prosecuted, rather than escaping with a caution.
But Mr Cameron, who says knife crime is a problem of "epidemic proportions" in the UK, says the presumption should go further - so anyone convicted of carrying a knife should be jailed.
Justice Secretary Jack Straw says ministers will review sentencing guidelines on knife crime when they come into force on 4 August.
Currently, about a third of those found with offensive weapons receive only a caution or final warning.
According to Home Office figures, 17% of people prosecuted for carrying a knife went to prison in 2006, compared with 6% in 1996. The average sentence length has increased by almost a third over the same period.