Most staff would get a £1,000 payment this year and a 4.3% rise next year.
The NEU has announced another two potential strike days in England and is urging members to reject the deal.
If they do, the decision will be passed back to the pay-review body, which recommended a 3% rise from September 2023, Ms Keegan says - and the £1,000 payment for this year would be lost.
The education secretary said she was disappointed unions were advising the rejection of "a fair and reasonable offer".
She said the government had gone "as far as we can" with negotiations and added that inflation is expected to be much lower next year, when the pay rise would take effect.
If the pay offer was accepted, "it will be funded, as we know that is important to schools", Ms Keegan added.
The government says it is giving schools £2.3bn over the next two years and starting salaries for new teachers will reach £30,000 next year.
But the National Education Union (NEU) is urging its members to vote to reject the latest pay offer because it is "not good enough" and "is not fully funded" - - meaning it would have to be funded partly by money already promised to schools.
A letter from Ms Keegan to the unions said that funding for the new pay offer would come from additional money awarded to schools in the Autumn budget and that schools will receive extra funding for the £1,000 one off payment and 0.5% of the pay increase for next year.
The result of their ballot will be announced on Monday, 3 April. And if they turn the deal down, the next two days of strike action in England will be:
* Thursday 27 April
* Tuesday 2 May
The NEU says it is speaking to head teachers to ensure exam preparation for GCSE and A-level students is not interrupted.
Three other unions, the NASUWT, Association of School and College Leaders and National Association of Head Teachers are also balloting members on the offer, with the NAHT also asking if they would take industrial action if it is rejected.
NAHT members first voted to strike in January - but turnout was 42%, below the legal requirement of 50%.