Meanwhile, they placed starter Drew Brees on injured reserve Friday afternoon -- meaning he must miss at least three games while recovering from cracked ribs and a punctured lung.
Hill took all the starter reps at practice this week, according to the source. A source also told ESPN's Dianna Russini that Winston will not be part of any offensive packages unless needed because of injury.
No timetable has been set for Brees' return. This is the first time he has been placed on IR, and this will be just the seventh game he has missed due to injury in his 20-year career.
The Saints also are signing veteran quarterback Trevor Siemian to their active roster to add depth at the position, according to Siemian's agent, Mike McCartney.
The Saints' decision to start Hill is a reminder that coach Sean Payton and team executives have always maintained that Hill could be a starting NFL quarterback. They backed up that belief by signing him to a two-year, $21 million contract extension as a restricted free agent this offseason.
Hill, who was a dual-threat quarterback at BYU from 2012 to 2016, has carved out a unique role with the Saints over the past four seasons as a read-option QB/RB/WR/TE/FB/special-teams specialist.
He has completed just 11 of 20 passes in his career, including the playoffs, for a total of 255 yards with no touchdowns and one interception. Hill has not thrown a touchdown pass since Nov. 26, 2016, in his final season at BYU.
However, the fast and physical 6-foot-2, 221-pounder has run the ball 105 times for 596 yards (5.7 yards per carry) with four touchdowns.
Hill played seven snaps at quarterback in the second half of last Sunday's win, running the ball on five of them for 31 yards. He lost a fumble on his final carry and was sacked once.
He has lost two fumbles this season and three in his career.
Payton declined to announce the Saints' starting quarterback after Friday's practice, despite being asked about the Hill reports.
"Someone has been named the starter, but we haven't announced it. Is that all right?" Payton said. "We'll do what's best relative to how we play the quarterbacks and the situations and what we're trying to accomplish to win this game."
The Saints did not make either quarterback available for media interviews this week.
It's unclear whether the Saints are committed to using Hill throughout Brees' injury absence or whether they plan to evaluate both of their backup quarterbacks at some point. But this time period will serve as a valuable evaluation for their future as well, because Brees, 41, could retire after this season.
The Saints signed Winston to a one-year deal worth $1.1 million plus incentives this offseason. Payton and teammates have credited Winston for his attitude, leadership and development throughout the season. He completed 6 of 10 passes for 63 yards last week with no touchdowns or turnovers, though he did throw one pass that was nearly intercepted.
Hill has become one of the league's most fascinating and polarizing players, since his ability to function as an NFL quarterback is still so unproven. Hill is already 30 years old; he went on a two-year church mission after high school and spent five years in college. He originally signed with the Green Bay Packers as an unrestricted free agent in 2017 before the Saints claimed him on waivers because Payton was so impressed by his preseason performance.
Hill finished his BYU career with 6,929 passing yards, 43 touchdown passes, 2,815 rushing yards and 32 rushing touchdowns -- despite suffering four season-ending injuries (knee in 2012, broken leg in 2014, torn ligaments in his foot in 2015, hyperextended elbow in 2016).
Hill initially broke out with the Saints in 2017 as a special-teams coverage ace, but his offensive role has expanded each year since then. His best performance came in January's playoff loss to the Minnesota Vikings, when he completed one pass for 50 yards, ran the ball four times for 50 yards and caught two passes for 25 yards and a touchdown.
This season, Hill is 4-of-5 passing for 86 yards with no touchdowns or interceptions. He has 34 carries for 186 yards and a touchdown and six catches for 74 yards and a touchdown.
Siemian, 28, has spent this season on the Tennessee Titans' practice squad after signing with the Titans in August. He went 13-11 as a starter with the Denver Broncos from 2016-2017 after being drafted in the seventh round in 2015. He spent the past two years as a backup with the Minnesota Vikings and New York Jets.