The three groups were shortlisted from 11 a week after reports surfaced of an investigation by the Serious Fraud Office for "suspected false accounting" and allegations of a £60m slush fund. BAE Systems denies any corporate wrongdoing.
The decision will be decided by a staff vote, and the two-week process started on Monday with a result due by the end of the month. The charities will pitch to staff at BAE's Preston site this week.
But the Campaign Against Arms Trade (CAAT) said that the money is tainted.
"Charities should not be linking with BAE because their support provides it with an element of legitimacy," said Ian Prichard, research co-ordinator at CAAT. "These charities have to realise that they don't exist in a vacuum.
How would they feel if another charity took money from a company causing damage in an area that they cared about?"
The Stroke Association's director of communications Greg Heines said: "This pitch is about BAE's employees rather than the work the company does."
Disability United, which comprises Action for Blind People, Leonard Cheshire, Mencap and the MS Society, was launched in May to pitch for corporate donations.
"This is exactly the opportunity that this consortium was set up to win," said Mark Bishop, group spokesman and head of corporate fundraising at Leonard Cheshire. "I'm very relaxed about the prospect of having BAE as a corporate partner."
The Wildlife Trusts head of partnerships Andrew Davis said: "We take the pragmatic view to working with companies. They are all part of society and embedded in many communities."
NCH, this year's partner, expects to receive £2m. It has also benefited from 4,000 staff volunteering days.
Simon Burne, director of marketing and fundraising at NCH, said the partnership is primarily a staff fundraising initiative, which met the charity's ethical guidelines.
A BAE Systems spokesman said no charities had turned down the chance to be considered for the partnership.