Hills followed up Saturday's 6-0 win over Belarus with an 8-0 thrashing of Turkey on Thursday at Fratton Park.
Toni Duggan scored a hat-trick as a crowd of 6,293 saw England's goal glut in Portsmouth, and there is a four-week gap before England's next World Cup qualifier, a home game against Wales at Millwall's Den home on October 26.
Hills, who spent 11 years working as Hope Powell's assistant, will now get the chance to stake his claim for the job on a permanent basis in the coming weeks.
FA director Kelly Simmons confirmed the process would soon be under way, and stressed the candidates were from a wide spectrum.
Simmons, director of the national game and women's football, said: "We certainly had a lot of global and domestic interest. It's a fantastic job. We're going to start interviewing very shortly and hope to make an announcement in the next few weeks so we'll go from there.
"There are male and female candidates, and domestic and international candidates as you'd expect. The game these days is global, coaches are moving across the world to coach in the top sides."
Powell was sacked following England's disappointing results at Euro 2013, where they finished at the foot of their group with just one point from three games after 15 years in charge.