The Times reports that he wants to strip the Ministry of Defence (MoD) of the Â£40bn project to build four new nuclear missile-carrying submarines.
He is said to have given an "ultimatum" that he will sign off the funds, but only if the Treasury has oversight.
The government said it did not comment on the contents of a "leaked document".
A spokesman said the government was committed to "maintaining a continuous at-sea deterrent", and more details would be provided at the Strategic Security and Defence Review, expected on 23 November.
A decision on whether to renew Trident - the UK's sea-based nuclear weapons system, made up of submarines, missiles and warheads - is due to be taken in 2016.
The Times quoted a "defence industry source" as saying there was a "tug of war" between the Treasury and the MoD, run by Defence Secretary Michael Fallon, over Trident renewal.
The chancellor is said to want the project to be given to a new body that reports to the Treasury, if he is to support the funding for it.
The unnamed source suggested Mr Osborne was concerned about the MoD's ability to deliver the new submarines on time.
"The MoD has a poor record on delivering submarines on time," said the source.
Asked about the reports, a government spokesman said: "While we are not prepared to comment on the contents of a leaked document, the government remains committed to maintaining a continuous at-sea deterrent, and to replacing the current Vanguard class of nuclear armed submarines with four Successor submarines."
"We will set out more detail with the publication of the Strategic Defence and Security Review."
The Prospect union, which represents defence specialists in the private and public sector, said it was "fanciful" to think the Treasury could take on the project.
"The idea that the Treasury - or indeed the Cabinet Office - has the skills to manage such a complex and technologically-challenging project as the Successor Programme is fanciful and needs to be fiercely challenged.
"Mr Osborne's land-grab will send shivers across Whitehall. Many will be wondering which major project will be next in his cross-hairs."
The Conservative government supports replacing Trident in its current form.
Labour has also been in favour, although leader Jeremy Corbyn opposes nuclear weapons and the party's policy is being reviewed.
Scottish Labour party delegates backed a vote to scrap the nuclear missile system, at the party's autumn conference.
The government has refused to confirm or deny reports that Chancellor George Osborne wants the Treasury to take charge of Trident renewal.