"If the House of Lords is to be relocated north we must use the power of architecture to express our political and economic ambitions," Foster said in a letter to British newspaper The Times.
To create that building Foster states that "an architectural competition, backed by a clear brief, would be the place to start".
He wrote the letter in response to plans by the UK government to explore the possibility of permanently moving the House of Lords from London to a city in the north of England.
It was reported last week that UK prime mister Boris Johnston has asked officials to investigate the practicalities of moving parliament's upper chamber outside of London, with York being the frontrunner.
"We should take our cue from the 19th century"
If the House of Lords is relocated, Foster believes that the UK should look to the past and especially the process that led to the creation of current home of the UK parliament – the Palace of Westminster.
"We should take our cue from the 19th century, which understood how great buildings could demonstrate confidence in our future," said his letter.
"After parliament burnt down in 1834, Britain applied its finest minds to creating a replacement. The process, overseen by Prince Albert as chairman of the Royal Fine Art Commission, gave us the magnificent Palace of Westminster."
The Palace of Westminster was designed by Charles Barry and Augustus Pugin following a debate between neoclassical and neogothic architects over the building's style. Foster is the current president of the Royal Fine Art Commission Trust.
"If we are to build a new House of Lords, we must set our sights every bit as high and produce work that represents the very best that our age can offer," he added.
House of Lords to be relocated as part of renovation
Plans are currently being developed to carry out a multi-year renovation of the Palace of Westminster, during which time both the House of Lords and the House of Commons will be relocated.
The current plan is for the House of Lords to be temporary based within the Queen Elizabeth II Conference Centre, which is located close to the Palace of Westminster.
A permanent plan to relocate the chamber is now being considered as part of a constitutional review that will also look at the role of the Supreme Court and the Lord Chancellor reported The Times.
According to the newspaper the government has already identified a site close to York railway station as a potential location for the House of Lords.
Since Parliament's plans to temporally relocate were announced many architects have suggested what this temporary home could look like.
Gensler proposed a floating bubble-like structure alongside the Palace of Westminster, while Foster suggested making exact replicas of both debating chambers within bubble-like structure behind Downing Street.