The 40-year-old had his first taste of English football in 1997 when he joined Bolton Wanderers from Finnish side Vaasan Palloseura for £100,000.
Jaaskelainen was released by Bolton in 2012 after notching up 529 appearances in all competitions over 15 years.
The Finland international, who was capped 65 times by his country, then joined West Ham where he played 61 times in three years, serving largely as a back-up to Adrian over the last 18 months.
His last Premier League appearance for West Ham was a 0-0 draw with Southampton in February.
Bradford’s only established goalkeeper is 32-year-old Ben Williams, who joined last year on a free transfer.