Scarlets veteran Jones has been rested for the clash with Samoa on Friday night, allowing Biggar the chance to stake his claim and impress Wales boss Warren Gatland.
The confident 20-year-old has already shaken off rival James Hook at the Ospreys and has now targetted Jones' No10 jersey.
"I feel I have established myself as first-choice number ten at the Ospreys and with Stephen here it's a massive challenge, but one I'm going to thrive on," said Biggar.
"Hopefully I will push Stephen all the way. I've got a chance on Friday evening and, hopefully, I'll take it and stake a claim for the next game.
"But, obviously I hope the end result is to take that 10 shirt in every game."
Biggar was left out of the squad for the narrow defeat to New Zealand last weekend, but the Ospreys ace aims to bounce back when he wins just his fourth Test cap against the South Sea Islanders in a 2011 World Cup dress rehearsal in Cardiff.
"I'd have loved to have started all the games at 10 this autumn but you have got to be realistic - Stephen is in pole position," added Biggar.
"It was the same at the Ospreys last season with James, so I know it won't happen overnight. But I want to be contention for the Argentina game next week.
"I feel if I continue playing the way I have during the opening three months of this season it will be a very difficult position for the coaches. I'd like to think it would be very close between Stephen and myself."
Scarlets star Jones will be 33 when the next global tournament comes around in New Zealand in less than two years' time, and Wales chief Gatland believes Biggar is heading in the right direction.
"The most noticeable thing about Dan is his confidence and maturity for a player of that age," explained Gatland.
"The way we want the game played and controlled, it's important we have a 10 that's quite vocal and domineering in that position.
"At the next World Cup, Stephen will be in his early 30's and could lose form and fitness. But it depends on how Dan goes on Friday and in the next year or two.
"He has a subtle kicking game and we don't want to coach the flair out of him. I believe he is going to have a positive impact on Welsh rugby."
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