The chair of the Hillsborough Family Support Group has said it would be a big mistake to hold a minute's silence in memory of Margaret Thatcher at football grounds this weekend.

The stance of Margaret Aspinall, who lost her 18-year-old son James in the disaster nearly 24 years ago, was backed by the Liverpool FC manager, Brendan Rodgers, amid a passionate debate about whether Thatcher's death should be marked at sports grounds.

The Premier League, the Football League and the Football Association have all said they have no plans to ask clubs to hold a minute's silence, despite calls from some former MPs and football chairmen.

The Wigan chairman, Dave Whelan, called for a minute's silence before his club's FA Cup semi-final at Wembley on Saturday, but it is understood the FA has no plans for such a gesture.

The football authorities privately argue that they are apolitical organisations and that Thatcher had no direct connection to the sport. There are also concerns about whether any silence would be honoured.

However, two Premiership rugby clubs – Saracens and Exeter – said on Wednesday that they planned to hold a minute's silence this weekend in memory of the former prime minister.

Reading FC, who play Liverpool on Saturday, two days before the Hillsborough anniversary, said they would hold a minute's silence for the 96 victims of the disaster. The club's chairman, Sir John Madejski, had earlier been one of those to back a minute's silence in remembrance of Thatcher, forcing it to clarify its position.

Aspinall said: "It's a terrible thing to speak ill of the dead, I know, because it happened to my son and all the Hillsborough victims for 23 years. But if they do hold a minute's silence for Margaret Thatcher, I think they are making a big mistake."

On Monday Aspinall will attend a memorial service at Anfield, the first since the Hillsborough independent panel revealed the scale of the establishment cover-up to deflect blame from the police and discredit fans.

"Somebody fed those lies, I think she was part of it and she knew about it. That's why I feel the way I do," said Aspinall, making clear she was speaking in a personal capacity rather than on behalf of the support group.

Sheila Coleman, of the Hillsborough Justice Campaign, said any silence for Thatcher would be "a huge mark of disrespect" to those who died.

Rodgers said the only reason to hold a minute's silence at this weekend's matches was to honour the victims of Hillsborough. "It is the only minute's silence there should be this weekend. We are entering a period of remembrance for 96 people who died going to a football game and for the families who have suffered for many, many years and have only just about got some sort of justification. That is the only remembrance there can be," he said.

The Saracens chief executive, Edward Griffiths, said the club's decision to hold a minute's silence in memory of Thatcher was not a matter of politics but "a question of respect".

"I don't think anyone would disagree that Baroness Thatcher made an immense contribution to this country and indeed the wider world. We think as a club that it is correct to respect that contribution," he said.

The Exeter chief executive, Tony Rowe, said he was not making a political statement in holding a minute's silence. "It is about showing a bit of respect. I lived through that era. Somebody had to stand up and be counted and she certainly did that, whether she was liked or disliked. I dread to think what sort of mess we would be in today if someone like didn't stand up."

The county cricket season began on Wednesday without any mark of remembrance for Thatcher, but clubs have been told to fly their flags at half mast next Wednesday on the day of her funeral.