However, the BBC may not be able to accept the offer from the Labour peer and multimillionaire businessman due to the corporation's strict rules on accepting donations for programmes.
Lord Hollick made a personal approach to BBC director general Mark Thompson last week at a charity art auction in Oxford, offering to pick up the shortfall to show the drama.
This followed a report that the BBC had shelved plans to reshow The Singing Detective on BBC Four this weekend to mark its 25th year after failing to agree a deal with the Potter estate.
Exact details of the dispute were not revealed, but the two parties were said to be apart by no more than £5,000.
On Twitter, Lord Hollick posted: "Bumped into Mark Thompson, BBC DG at Art Room charity auction and confirmed my £5k gift to help BBC fund rebroadcast of Singing Detective."
The BBC had intended to show the landmark series, starring Michael Gambon, from this Sunday, but is understood to have baulked at the Potter estate's demand for £20,000 in repeat fees, reports The Guardian.
On BBC insider told the newspaper: "The BBC is understood to be considering ways that could make it possible for Hollick to make the unprecedented offer, although it is unclear whether it can directly take Hollick's money, as it is 'unable to accept donations in exchange for programmes'."
A possible solution would be for Lord Hollick to donate the £5,000 directly to the Potter estate on the proviso that The Singing Detective would be reshown on the BBC.
In a statement, a BBC spokeswoman said: "It would be inappropriate to pay above the odds for any programme, particularly during a time of budget cuts."
The Singing Detective caused a stir when it was aired by the BBC from November 16 to December 21 in 1986, and has proved hugely influential on TV ever since. In 2003, the drama was adapted into a movie starring Robert Downey Jr and Mel Gibson, with the locations changed to the US.