Their suspicion is understandable. The tabloid press has generally been destructive rather than helpful to Labour leaders. Neil Kinnock and Michael Foot, in particular, bear the scars of vicious attacks. But their party has not been alone in facing such treatment. When Beaverbrook and Rothermere attempted to unseat Stanley Baldwin in 1931, the Tory leader condemned the press for exercising 'power without responsibility - the prerogative of the harlot throughout the ages'.
John Major may share his predecessor's views.
Given recent history, a distrustful Labour Party would prefer to make its choice free of much debate beyond its own confines. Members would perhaps like an election more akin to that which gave the succession to Mr Smith in 1992. They will be disappointed. This contest will rightly be conducted in the full glare of publicity because the winner stands a good chance of becoming Britain's next prime minister.
In contrast, the election of Mr Smith seemed of limited relevance to politics at the time, given the despair of his party, the freshness of the fourth Tory victory and the distance of the next general election. Furthermore, increased democracy within the party has now turned the campaign into an event more like a US presidential primary.
Tory leadership elections increasingly have a similar flavour. Even though only MPs are entitled to vote, these contests are still a national media event. During the battle to take up Margaret Thatcher's mantle, Douglas Hurd, an Old Etonian, spent a great deal of time playing down his class origins. This tactic was surely aimed more at the public than at fellow MPs.
As far as Mr Blair is concerned, the Tory press - along with non-partisan elements of the media - seem genuinely impressed by him. This partly reflects disillusionment with the Conservative Party and particularly with Mr Major. Some newspapers are perhaps preparing for the real possibility that Labour could win the next election. They want a PM they can live with and one more sympathetic to their southern readers than was Mr Smith.
If malice is emerging, it is evident in mischievous claims of a party backlash against Mr Blair on the grounds that he is too popular with the media. This is nonsense. The fact that not everyone wants to crown Mr Blair tomorrow is a mark of healthy democratic debate. The nature of the Labour Party meant there was always going to be support for a candidate of the left. That does not constitute a backlash.Reuse content