David Moyes's side will also take on Leyton Orient on Tuesday, meaning that the Hammers will split their squad in two for the double header, with both matches kicking off at 7pm.
Northampton kicked off their pre-season schedule on Saturday afternoon with an impressive 6-0 victory over Spartan South Midlands League Division One side Sileby Rangers at Fernie Fields.
Summer arrivals Jonny Maxted, Paul Lewis, Nicke Kabamba, Dylan Connolly and Sid Nelson were all involved in the team's first friendly of the summer, and Kabamba was the star of the show with a brace on his debut.
Northampton are back down in League Two, having been relegated from League One last season, and they will be led into the new campaign by Jon Brady, who was appointed permanent manager in May.
The Cobblers have an incredibly busy run of matches coming up, facing Stamford on July 17, Nottingham Forest on July 20, AFC Rushden & Diamonds on July 21, Birmingham City on July 24, Brackley Town on July 27 and Cambridge United on July 31.
Northampton will then open their 2021-22 League Two campaign against Port Vale on August 7, and Brady will hope to have more new players on board before the start of competitive action.
Moyes's side, as mentioned, will also take on Leyton Orient on Tuesday evening, and it is currently unclear which group of players will be selected for which fixture.
West Ham, who finished sixth in the Premier League last season to qualify for the 2021-22 Europa League, will have their full squad back together for the pre-season clash with Reading on July 21.
The Hammers will then take on Celtic on July 24 and are expected to play at least one more friendly between the match against the Scottish giants and their Premier League opener at Newcastle United on August 15.
It has been a quiet transfer window thus far for the London club, but there are expected to be a few incomings and outings before the deadline passes.
Northampton had two trialists in their first XI against Sileby Rangers in the shape of Isaiah Bazeley-Graham and Josh Debayo, and it will be interesting to see whether they are involved in this match.
Summer arrivals Kabamba and Pinnock were among the goals for the Cobblers last time out, and all five recent signings could again take to the field from the first whistle here.
Head coach Brady named a different starting XI in each half and that is again expected to be the case on Tuesday, but Jordan Flores, Caleb Chukwuemeka, Joseph Mills and Max Dyche will have to be assessed, having missed out on their side's first pre-season fixture through a mixture of injury and self-isolation.
The fact that the squad is being split in two means that the players should get more minutes in their legs, and it will be interesting to see whether Mark Noble features here or against the O's.
Head To Head
Tuesday's match will be the 34th meeting between these two teams in all competitions, and interestingly, Northampton actually lead the head-to-head record 14 victories to West Ham's 11.
The last league meeting came back in January 1966, when the pair played out a 1-1 draw, while 1-1 was also the score when the two teams last locked horns in the Football League Trophy in November 2016.
We say: Northampton Town 0-2 West Ham United
It is difficult to know exactly what to expect from West Ham considering that their squad will be split in half for two fixtures on the same night. That said, the Hammers are expected to have a mixture of youth and experience on show here, and the Premier League side should have too much quality for their opponents.