"We had the financial muscle, but the risk reward calculation wasn't clear," he told Goal.
"Spending a lot of money on a player that didn't work out would have really put the club in some financial straits.
"Did we miss out on talents? Certainly. I remember sitting in meetings with Steve [Rowley], Ivan [Gazidis] and Arsene [Wenger] and gnashing our teeth over Thibaut Courtois. We knew he was good.
"Eden Hazard as well, we wanted to do that deal. But there was a sense of responsibility and running the club prudently, all the time knowing that every £1 we spent we had to act as if it was our money. It was very, very difficult."
Law expressed frustration at the situation but admitted there wasn't much Arsenal could do given their financial position.
"We always felt that we didn't have the option to make a mistake because a £10m-£20m mistake would be a serious one for the club.
"Where United would take risks on players, we just didn't have that luxury. We couldn't take those risks."
Hazard and Courtois are now together at Real Madrid.