"We have got to find a minimum of five wins...between now and the end of the season to be safe," he told a news conference on Thursday.
"We've won three, drawn two and lost two in our last seven, which has been the best run since I've been here."
Allardyce took over at West Ham when they were relegated in 2011 and won immediate promotion back to the top flight.
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He left the Londoners at the end of last season when his contract expired, amid complaints from some fans about the team's playing style.
Slaven Bilic, manager of seventh-placed West Ham, has praised Allardyce for the work he did at Upton Park and believes Sunderland can avoid the drop.
"Sam got us up straight away and stabilised us in the Premier League so he did a great job here," said Bilic.
"Sunderland are very organised, they are showing not only the quality but the character. There is still a long way to go for them but they have a big chance to stay in the league."
(Reporting by Simon Jennings in Bengaluru; Editing by Tony Jimenez)
