Celtic are more than halfway to achieving the first unbeaten season in the top flight since the 19th century and are only three clean sheets away from beating a club record of 10 consecutive shut-outs set in 1914 and equalled in 1922.
They are also looking for their 11th consecutive Scottish Premiership win when they face Hibernian at Easter Road on Sunday, although they are some way off their own British record of 25, which they achieved under Martin O'Neill in the 2003-04 season.
But Mulgrew is more focused on winning the game in front of him than making history.
The Scotland international told Celtic TV: "It doesn't get spoken about much to be honest. I think everybody knows in the back of their mind. But it's the old cliche of one game at a time.
"We don't speak about clean sheet records or winning runs too much.
"We look to win the next game we're in and if that means beating records then so be it."