The late loss, sealed by Lyle Foster's 95th-minute strike, stretched Wolves' winless run to nine league matches and left them six points adrift of safety.
However, club sources have indicated that Pereira's position is not under immediate threat, with the board maintaining faith in the Portuguese coach who signed a new three-year deal in September.
Speaking after Sunday's defeat, Pereira admitted results have been hard to take but insisted he remains focused on turning things around.
"It's normal in football. If we win two or three games in a row, everything changes," Pereira said. "Two months ago, they sang my name for the work we did last season to stay in the Premier League. Now maybe they sing my name to sack me.
"I'm not worried about my job. It's about helping the players stay mentally strong for the next fight."
Wolves were undone by a combination of missed chances and defensive lapses against newly-promoted Burnley, who claimed their first away league victory of the season. Zian Flemming struck twice in the first half, converting from Quilindschy Hartman's crosses, before Wolves fought back through Jorgen Strand Larsen's penalty and Marshall Munetsi's header just before the break.
Both sides pushed for a late winner, but Burnley held their nerve as Hannibal Mejbri slipped Foster through on goal in stoppage time to snatch all three points.
Pereira now faces a crucial week that could shape his future. Wolves meet Chelsea in the Carabao Cup on Wednesday before a Premier League trip to Fulham at the start of November, followed by another clash with Chelsea in the league.
A first victory under Pereira could calm the storm, but another setback may intensify the pressure on the Wolves boss as his side fight to avoid being cut adrift at the bottom.
Nottingham Forest are set to receive a double injury boost ahead of their Premier League showdown with champions Liverpool after the international break, with both Chris Wood and Callum Hudson-Odoi expected to return to action.
David Moyes has praised James Garner for his outstanding form this season, saying the Everton midfielder is edging closer to an England call-up.
Leeds United manager Daniel Farke has provided a detailed update on the club's injury situation as the Whites head into the international break aiming to reset after a frustrating run of results.
Arne Slot believes Liverpool's toughest spell of the season is now behind them, with a crucial run of winnable fixtures offering a chance to reignite their campaign after a heavy defeat to Manchester City.
