Thiago, 25, established himself as one of the division's most prolific forwards last season, scoring 22 goals and adding an assist in 38 league appearances, second only to Erling Haaland in the Premier League's scoring charts.
The Brazil international earned his first senior caps as a result and was included in his country's World Cup squad this summer.
According to journalist Rudy Galetti, Nottingham Forest have now turned interest into a concrete offer, submitting a £70 million bid despite Thiago being contracted to Brentford until 2031 following a new long-term deal signed last season.
Arsenal, Manchester United, Chelsea and Bayern Munich have all been credited with monitoring the forward's situation this summer, though none of those clubs have yet moved to match Forest's bid.
Thiago arrived at Brentford from Club Brugge in the summer of 2024 and endured an injury-hit start to his time in west London before establishing himself last season.
Brentford, who opened their own campaign with a 3-0 win over Tottenham Hotspur that saw Thiago play 83 minutes, have given no indication of their response to Forest's offer, though reports suggest a late departure before the transfer window closes cannot be ruled out given the scale of interest in the striker.