It said it would raise about £150m.
It added that the share sale would support its plans for growth and further improve its ability to recruit and retain top staff.
Once the company is listed, it will pay £50m to the Treasury to settle what it still owes after buying  Northern Rock plc in late 2011.
Virgin Money chief executive Jayne-Anne Gadhia said each employee would receive £1,000 worth of shares in the business when the flotation was completed later this month.
Ms Gadhia told the BBC's business editor, Kamal Ahmed, that she believed taxpayers had gained all their money back from the bailout of Northern Rock.
"In 2007, it looked like the taxpayer was going to make a big loss," she said. "Now this is the first bank properly out of the financial crisis.
"Debts to the taxpayer have in every sense been fully repaid. The taxpayer has got their money back."
Sir Richard said: "This is a huge day for Virgin Money. We started this company 20 years ago with Jayne-Anne Gadhia when we set out to challenge the financial services industry.
"Our wonderful team have come a long way since then and have built a strong and valuable business offering great value products and services and a real challenge to the established players."
Analysis: Kamal Ahmed, business editor, BBC News
Virgin Money's announcement of its intention to sell a stake on the London Stock Exchange brings to an end another chapter in the sorry story of Northern Rock.
The functioning bits of the bank, which spectacularly collapsed in 2007 and was bailed out in February 2008 with £1.4bn of taxpayers' money, were bought by Sir Richard Branson and US investor WL Ross for £747m in 2011. With subsequent payments, that figure has risen to £1bn.
Further value is left in the "bad bit" of the bank, which was hived off under the UK Asset Resolution scheme, which is now returning profits to the Treasury.
So has the government got its money back? Probably, although it is very difficult to quantify the actual costs of the multiple bailouts, of which Northern Rock was only one.
Virgin Money has 75 branches throughout the UK and has 2.8 million customers.
It employs 2,800 people, of whom 1,800 work at its centre of operations in Gosforth.
In November 2011, Virgin Money bought the banking and mortgage lending arm of the old Northern Rock bank, which was bailed out by the Bank of England in the autumn of 2007 at the start of the international banking crisis.
Northern Rock was nationalised in early 2008 and then in 2010 it was split in two - Northern Rock plc  and Northern Rock (Asset Management), into which was placed its bad debt.
Virgin Money's final £50m payment will take the amount it has paid the Treasury for Northern Rock to £1.02bn.
Last week, another UK challenger bank, Aldermore, announced its own plans to float this month, saying it hoped to raise £75m.
While Virgin Money is a retail-only bank, Aldermore, founded in May 2009, is a specialist bank that concentrates on lending to small and medium-sized businesses, as well as to homeowners.

The 30-year-old made one appearance for Vale, in their 2-1 defeat by Carlisle in the EFL Cup first round on 9 August.
"Having moved to England from Portugal over the summer, the striker has found it difficult to adjust to life in the country," a club statement said.
The former Sporting Lisbon player will now return to Portugal.

The South Africa-born 30-year-old claimed 6-66 on his debut against Nottinghamshire at Old Trafford.
Wagner, who had a stint with Northants in 2014, joined the Red Rose as their overseas player for the 2016 campaign.
"It's a great club and a great place. I always had something for Lancashire when I played here for Ormskirk in 2008," he told BBC Radio Lancashire.
"I came to watch quite a few games for Lancashire and when England played here as well, so I love the ground and everything about it.
"For me to come over here was an easy decision and to come and play a part in such a special club is pretty awesome. I'm lucky to be here."
Wagner's figures were the best by a bowler making his Lancashire first-class debut since 1923.
Virgin Money, the UK challenger bank part-owned by entrepreneur Sir Richard Branson, has announced plans to float shares on the London Stock Exchange.
[NEXT_CONCEPT]
Port Vale have terminated the contract of Portuguese striker Carlos Saleiro, just six weeks after he signed for the League One club.
[NEXT_CONCEPT]
New Zealand seamer Neil Wagner has spoken of his pride at being able to play for Lancashire this season.