Toney was left out of England’s World Cup squad last week and he has subsequently been charged by the FA with 232 alleged breaches of their betting rules.
The striker had been tipped for a call-up to the World Cup squad after scoring 10 Premier League goals, a tally that is only beaten by Erling Haaland and Harry Kane.
It has been speculated that Toney being investigated, and then subsequently charged by the FA, impacted his World Cup chances.
But asked if he told England manager Gareth Southgate not to select Toney, Bullingham said to the BBC: “No, I did not say that to Gareth.
“He was absolutely available for selection and he was not chosen on footballing grounds.”
Toney has been charged with the alleged breaches over a four-period between 2017 and 2021 and, if found guilty, he could face a long ban. The 26-year-old striker has been given until next Thursday to respond to the charge.
Brentford said in a statement: “The club has been in discussion with Ivan and his legal representatives about this matter and those conversations will continue privately. We will make no further comment until the matter has been completed.”
The punishment for breaching FA betting rules varies from case to case, depending on a range of factors, including the number of bets and how much was staked.
If he is found to have placed bets on his own side to win, he could potentially face a six-month ban.
A statement from the FA read: “Ivan Toney has been charged with misconduct in relation to alleged breaches of the FA’s betting rules. [He] has until Thursday, November 24, to provide a response.”