Tory Justice Secretary Chris Grayling wants to impose a mandatory six-month jail term for any adult convicted of a second offence involving a knife.
Changes to the law are backed by other high-profile Tories including Home Secretary Theresa May and Education Secretary Michael Gove.
But senior Lib Dems argue sentencing should be left to judges.
In a leaked letter to the Daily Mail, the Lib Dem Chief Secretary to the Treasury Danny Alexander said jailing more offenders caught with knives would be too expensive.
In the letter to the Deputy Prime Minister Nick Clegg, Mr Alexander said: “It would impose a substantial new pressure on the prison population… this proposal would certainly not be affordable before 2016-17.”
In another letter to Mr Clegg, Lib Dem Schools Minister David Laws said he would not not back mandatory minimum sentences.
Mr Laws wrote that it should be for judges to decide on what sentences to impose on repeat offenders.
He said: “I have concerns about the impact on the prison population and the deliverability of the policy within this Parliament.
“I am therefore unwilling to clear this proposal.”
The coalition split on the issue comes at a particularly sensitive time following the stabbing of teacher Ann Maguire in a classroom in Leeds.
Mr Gove told Sky News: “I think it’s really important that we send a very clear signal that knife crime is going to be dealt with effectively.
“I think it’s absolutely important that we use Parliament to communicate to the public, and to anyone tempted to carry a knife in public, that the sentence for behaving in this way will be clear, and firm, and tough.”
He said he had been “deeply moved” by the death of head teacher Philip Lawrence, who was murdered outside a school in Maida Vale, London, in 1995.
Mr Gove said: “Ever since then I have been trying to ensure that we both change the law, in order to deal more effectively with knife crime, and that we tackle some of the underlying problems that lie behind this scourge.”
A Lib Dem source said: “The idea that Nick Clegg, Danny Alexander or any other Lib Dem would wish to make our society less safe is ridiculous.
“Those people who have leaked this correspondence out of context and those seeking to exploit it on a Party basis in such a tragic week will have to examine their own conscience.
“The Liberal Democrats are interested in what actually works to tackle knife crime, rather than tough sounding gimmicks.”
The move to impose a mandatory prison sentence for adults convicted of a second knife offence was first raised by Mr Grayling in 2012.
An MoJ spokesman said: “Tough sentencing options for knife crime are available, but we aren’t complacent, and we continue to keep this area of the law under close scrutiny.”