The Tate Modern, the Lowry Centre, the Eden Centre and The Deep are on a shortlist of 60 lottery-funded buildings to which the Millennium Commission is preparing to give money.
The lottery's commissioners, which include former deputy prime minister Michael Heseltine and children's television presenter Floella Benjamin, are meeting to decide the amounts.
The Tate Modern, in central London, which opened to the public in May 2000, is thought to have requested £5m for refurbishments.
A Tate spokesperson said: "The Tate Modern has applied for funding to make improvements that will cater for more than double the number of customers than was expected."
The Lowry in Salford, Greater Manchester, is believed to have asked for £1-5m to convert space into the "ArtWorks@TheLowry".
This development will include classrooms, a studio theatre and conference facilities.
In its submission, the Lowry said: "It will contribute to the Lowry's economic wellbeing through the provision of additional corporate hospitality facilities, matching demand for capacity at the venue."
Lowry property and construction director Steve Thorncroft said: "We are delighted to have been shortlisted and are extremely optimistic, as we believe it to be a very strong and exciting scheme."
The Eden project in Cornwall is looking for lottery money to top up the private funding it requires to build the £80m third phase. This will include an education centre and a 1000 m2 desert environment.
The Deep aquarium in Hull has asked for £2.4m to go towards its planned £6.5m, 2000 m2 extension. Colin Brown, the chief executive of EMIH, the company that runs the aquarium, said: "We are unable to cope with demand at busy times, so we feel that the extension will make something that is already pretty good world-leading."
The Millennium Commission is funding the projects largely through money that it held back in the late 1990s, when it underestimated the amount of money the lottery would generate. A spokesperson for the commission said that the exact amounts it would be able to give to bidders was uncertain because of interest-rate fluctuations.
But Millennium Commissioner Judith Donovan said: "The commissioners can give out a penny or £100m to a project. This has involved paperwork the size of a forest."
The commission was originally due to decide on the funding for the projects in May but the unexpectedly large number of applications, thought to be nearly 200, caused a delay.
It handed more than £1m in awards to about 100 community projects in May. Last month it gave the Peterborough Millennium Green Wheel £490,788 and Our Dynamic Earth in Edinburgh £263,500.