The petition, which has garnered 8,361 signatures, was submitted by Westmorland and Lonsdale MP Tim Farron.
It opposes the plans and calls on the Government and health bosses to ensure residents in the South Lakes don’t have to travel any further for acute hospital treatment.
Over the last few months, Mr Farron has raised this issue with the Secretary of State for Health in the House of Commons and met with Health Minister Edward Argar to warn him of the "devastating consequences the plans will have for local patients".
He said: "It’s all very well having a brand spanking new hospital, but not if it’s so far away that local people don’t arrive there alive.
“It’s vital that A&E is as close to people as it can be, and certainly no further away than it already is.
“Thank you so much to everyone who signed this petition – we won’t stop fighting until this plan is off the table.”