Molinari and Barham stand at 16-under-par 200, and both lead a European Tour event heading into the final round for the first time in their careers. Their advantage is one shot over Philip Archer, who carded a 4-under 68 to reach minus-15.
Nicolas Colsaerts fired the low round of the day -- an 8-under 64 -- to vault from 38th place overnight into a tie for fourth. He was joined at 14 under by first-round leader Soren Kjeldsen (69) and Benoit Teilleria (65).
Molinari and Barham were both tied for second place overnight behind David Drysdale, who was the only player currently sitting in the top 27 who did not shoot a round under par on Saturday. Drysdale stumbled to a 1-over 73 and is tied for 10th place at minus-12.
Cooler weather and some rain is in the forecast for Sunday, so the scores may not be as low as they have been on the first three days. But one player is likely to receive plenty of warmth during his final round.
Playing in the final group in his home country, Molinari will be seeking his first professional title on Sunday and the first win for an Italian at this event since Massimo Mannelli's in 1980.
The gallery on Saturday followed Molinari -- brother of reigning U.S. Amateur Champion Edoardo, who missed the cut in last place -- enthusiastically.
'The crowd made a little difference,' Francesco Molinari said. 'I am happy to be here and have the support of the crowd. Hopefully we can do it tomorrow.'
Molinari's third-round performance mirrored those of his first two rounds: a steady mix of pars and birdies, and a third straight round in the 60s.
'I think [winning] would be great for Italy and for my family,' said Molinari, who is a year younger than his amateur brother. 'My brother has not played so well this week, but I am sure that he will join me on tour next season.'
Molinari began his third round with a birdie on the first hole, a par-5, but didn't start making a move until the back nine. After that first birdie, 10 consecutive pars followed before the 23-year-old went back-to-back with birdies on 12 and 13 to reach 14 under.
After another par at 14, Molinari birdied the par-5 15th and the par-3 16th to get to minus-16. He hasn't bogeyed a hole since the first round.
'It was important to make the birdie on the 12th because if I had kept making pars then I may have dropped a shot eventually, so it was good to make the birdie and start a run,' he said.
Barham, 29, has been nearly as steady as his co-leader through the first three rounds. After opening with a birdie at the first hole Saturday, he slipped to just his second bogey of the tournament at the par-4 second.
Three more birdies on the front nine put Barham at 14 under around the turn, ahead of his eventual co-leader. He then added birdies at the par-5 12th and the par-4 18th, his first birdies at either hole this week, to reach 16 under.
Barham's final birdie came on a 35-foot putt -- the type of clutch stroke that may come with added pressure Sunday in front of a pro-Molinari crowd.
'I have thought about the crowd, but they are actually very good,' said Barham. 'I think as long as they see lots of good golf they are not fussed who it's from.
'I have been just trying to remember all the good shots that I have hit rather than reflect on the bad ones and I think that's important. You can't go round the course trying to think about technique all the time, you have to concentrate on getting the ball in the hole.'
Sharing seventh place at 13 under were Miguel Carballo (66), Peter Fowler (68) and Anders Hansen (66). Drysdale was tied one stroke further back with Bradley Dredge (68), Richard Green (68), Hennie Otto (67) and Jarmo Sandelin (67).