Current boss Arsene Wenger is reported to have around £90m to spend in the next few weeks as the Gunners aim to win their first trophy since 2005.
"They need three or four quality players that can go straight into the first team," Graham told talkSPORT. "They have got quite a lot of money so it as an important summer for Arsenal and Arsene Wenger.
"They need an experienced second-choice goalkeeper, a top-quality physical centre-half, a holding midfield player and a finisher up front."
Arsenal start the new Premier League campaign at home to Aston Villa on August 17.