The pair would need to be signed and registered by midday on Friday to be able to play in the Londond derby at Emirates Stadium.
Arsenal have agreed fees for both players, which are worth a combined total of over £50million, and they are now working through medicals and personal terms, with neither expected to be an issue.
The Gunners have agreed to pay Sheffield United £24m for Ramsdale, although that fee could rise by a further £6m through add-ons.
The add-ons are performance-related and will only be triggered if the England goalkeeper makes numerous first-team appearances and Arsenal qualify for Europe multiple times.
Sheffield United had initially been demanding £40m for Ramsdale, but a compromise was struck after the goalkeeper expressed his desire to move to north London.
“We need to evaluate the situation. We need to check what is best for us now, now that we have lost Aaron,” Sheffield United boss Slavisa Jokanovic told The Star.
“Listen, I will repeat one more time, it’s not what I wanted. It is difficult to stop this kind of situation and it’s a huge deal for the club, after being relegated from the Premier League. From my side, he was our best option and we wanted to be working with our best players.
“Aaron was our first option and we are now looking at what is the best step for us. It was not perfect preparation to lose Aaron the day before the game but we can not say we were surprised by it.”
A deal for Odegaard is also close to completion, with Arsenal set to pay in the region of £30m for the attacking midfielder. Odegaard is in London to undergo his medical and sign for the Gunners, where he spent last season on loan.
Arsenal have been monitoring the Norwegian’s situation all summer, but originally Real Madrid had shown a reluctance to sell. Their stance, however, softened last week and Arsenal have acted swiftly to bring Odegaard back to the club.
The 22-year-old enjoyed his loan spell at the Gunners last season and he is keen to work under manager Mikel Arteta again.
Odegaard is also understood to be attracted to the idea of regular first-team football, while he felt settled living in London too.