Technically, it is possible to run Eurostar trains to Manchester. When the channel tunnel was planned it was envisaged Eurostar trains would run on the East and West coast main lines, and Britsh Rail spent £140 million on track works to make this possible. A Eurostar depot was also built in Longsight, but it remained empty for years.
Trial runs were undertaken with Eurostar trains, but the project was shelved, mainly due to the projected passenger numbers being lower than first thought. Numerous train companies offered to run services, but they would have required a subsidy which the government refused to sanction.
So, to answer your question, it is possible to run Eurostar trains from Manchester to Paris.