The German has started to be questioned for the first time in his Reds reign following an indifferent run of results.
Having entered the first international break of 2017-18 on the back of four successive victories, Liverpool stumbled into the second having collected just one win in seven.
Klopp has seen his methods and transfer business placed under the microscope, but a former chief executive at Anfield believes there is no better man to challenge the elite on a limited budget.
Purslow told Sky Sports: “He’s two years into this job that looked to be perfect for Jurgen Klopp.
“There are great similarities between what he achieved at Dortmund, what he was expected to achieve - which was to compete with an absolute behemoth in Bayern Munich, while spending less money, both on wages and in the transfer market.
“That is the job description at Liverpool.”
Purslow added on Liverpool having to compete with big spenders such as Manchester United, Chelsea and Manchester City: “There are revenue differences between Manchester United and Liverpool. Liverpool’s are still superb but they are a fraction of Manchester United.
“And yet the Liverpool manager is expected to compete with United and beat them, exactly as Klopp did - winning two league titles in Germany against Bayern Munich with a fraction of their spending.
“He is perfect for Liverpool. It is an innately difficult project, to compete with less money.”
Klopp will get a chance to prove just how close Liverpool’s class of 2017-18 are to United on Saturday when the Red Devils visit Anfield.
The Reds finished two places and seven points clear of their old adversaries last season, but have slipped seven points adrift of the early Premier League pacesetters this time around.
Criticism continues to be levelled at Liverpool’s defensive unit, with an inability to collect clean sheets having cost them domestically and in Champions League competition.
Things are not about to get any easier either, with Klopp’s side set to travel to Maribor for a crucial European encounter and Wembley to take on title hopefuls Tottenham after playing host to United.