The Competition and Markets Authority (CMA) said it was worried the tie-up could increase delivery prices.
The regulator gave the two firms five days to address its concerns about competition in the food delivery space.
If they fail to impress the watchdog, the $575m (£437m) deal could ultimately be blocked.
The CMA said that the investment may prevent Amazon from launching its own food delivery operation, a move which could reduce prices for customers in the future.
“Millions of people in the UK use online food platforms for takeaways, and more than ever are making use of similar services for the same-day delivery of groceries,” said Andrea Gomes da Silva, executive director of the CMA.
“There are relatively few players in these markets, so we’re concerned that Amazon having this kind of influence over Deliveroo could dampen the emerging competition between the two businesses.”
She said that if the deal were to go ahead in its current form then there was a “real risk” that customers, restaurants and grocers would face higher prices and lower quality services.
In a statement, Deliveroo said it was confident it could persuade the CMA that Amazon’s investment would “add to competition, helping restaurants to grow their businesses, creating more work for riders, and increasing choice for customers”.
Amazon said the investment would help Deliveroo develop new products and remain competitive.
It is not the first time Amazon has tried to enter the food delivery market in the UK.
The online retailer briefly ran its own delivery venture, Amazon Restaurants UK, which it started in 2016 but closed just two years later.
It was previously reported to have made approaches to buy Deliveroo outright. Uber also reportedly had talks with Deliveroo regarding an interest in buying it.
Deliveroo founder and chief executive Will Shu came up with the idea for the firm after he moved from New York to London as a banking analyst. He was working long hours and was frustrated by the fact so few restaurants delivered, a service widely available in the US.
In the firm’s early days, Mr Shu delivered all the food himself on a motorbike, while Greg Orlowski, his co-founder who has since left the business, developed the booking technology from his home in the US. Mr Shu still claims to get on his bike once a week to deliver an order to customers in London, as a way of staying in touch with riders.
Deliveroo has grown quickly – last year it doubled the number of towns and cities it operates in, helping sales to surge by 72%.
However, losses deepened to £232m in 2018, from £199m a year earlier as the company splashed out £18m on technology and £10m on free insurance for its riders.
Sales grew to £476m in the year to December 2018, up from £277m in 2017.
Deliveroo operates in 11 markets outside the UK including Australia, France, Germany and Hong Kong.