Trevor Muirhead, 44, and Neil McKenzie, 43, were jailed for five years in April last year for conspiring to assault Lennon, former MSP Trish Godman and the late QC Paul McBride by sending devices they believed were capable of exploding and causing severe injury.
McKenzie, from Saltcoats, and Muirhead, from Kilwinning, both in Ayrshire, tried to have their convictions quashed at the Court of Criminal Appeal in Edinburgh, where a hearing in the case was held in December.
Their legal teams argued there was insufficient evidence to allow the jury at the trial to find that the pair believed the packages were capable of exploding.
But three senior appeal judges ruled that both convictions should stand.
As reported in later editions of yesterday's Evening Times, judge Lord Menzies told the court during a brief hearing: "These appeals are refused."
The prosecution's case was that, even though none of the devices was viable, Muirhead and McKenzie believed they were capable of exploding.
Both men were originally accused of the more serious charge of conspiring to murder their targets, but it was thrown out due to insufficient evidence a day before the trial concluded at the High Court in Glasgow.