The party’s city manifesto, which will be launched this week, will include a commitment to develop two new spurs of the existing line.
Adam McVey, leader of the SNP group in Edinburgh Council, insisted it was time to turn the tram line “into a network worthy of a modern European capital”.
But critics insisted the move would “only deliver more misery” for the city.
Edinburgh’s trams launched in 2014, three years late and massively over budget.
The project was hugely controversial and a long-running inquiry into what went wrong is yet to reach a conclusion.
One of the proposed new lines will lead northward from the city center to Granton, where 3,500 homes are being built as part of a £1 billion Granton waterfront regeneration.
The second will head south-east towards the Royal Infirmary and the BioQuarter research district, and potentially beyond.
The SNP said current construction of the line to Newhaven remains on budget and on schedule to open for passengers in spring 2023.
It will add eight new tram stops and bring services into some of the most densely populated parts of Edinburgh.
However, the line will still be shorter than similar systems in Dublin and Manchester, as well as those of many cities across Europe.
Mr McVey said: “By keeping the Newhaven extension on time and on budget we’ve shown that, under the SNP, tram works can be delivered well.
“Even the pandemic hasn’t knocked the project off course.
“Now it’s time to turn Edinburgh’s tram line into a network worthy of a modern European capital.
“We will take the time to plan this properly, just as we did for the Newhaven extension, to minimize disruption as far as possible and ensure the new routes are deliverable.
“To tackle climate change, we have to provide people with low carbon alternatives for getting about the city. Trams are a proven way of doing that.
“Linking up parts of the city with fast, efficient transport supports residents to switch from car to public transport for those journeys.
“When the trams started running they carried millions of passengers – and bus passenger numbers also increased.”
Edinburgh is currently run by an SNP/Labour coalition.
Scottish Conservative council candidate Iain Whyte said: “People living in Edinburgh will be frankly astonished that the SNP are talking up more tram lines.
“The SNP’s track record on trams is nothing short of embarrassing yet they are asking voters to trust them to deliver this.
“The SNP council are currently claiming that current works are on time and on budget, but they haven’t delivered.
“Passenger numbers simply aren’t there, yet the SNP want to chuck more money onto the project.
“This is continuing to let down residents and businesses who have had to endure endless delays while works have taken place.
“It also signals quite a remarkable about turn, given the SNP wanted to scrap the trams when they got into office.
“Whether it’s trams, ferries or trains, you just can’t trust the SNP when it comes to transport infrastructure. These sort of plans will only deliver more misery for our capital.”