The 21-year-old is set to become the Gunners' highest-paid player after a breakthrough in talks.
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It is understood Saka will sign a five-year deal worth £300,000-a-week to reward his exceptional season.
The England forward has 12 goals and 10 assists in the Premier League this season, helping to inspire Arsenal's unlikely title charge.
However, some Gunners supporters will be concerned about history repeating itself.
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Big-money contracts for Mesut Ozil and Pierre-Emerick Aubameyang have backfired in recent years - but Woodsy insists that won't happen with Saka.
The presenter told the talkSPORT Breakfast show: "Personally, as an Arsenal fan… we had [Liverpool legend] John Barnes on yesterday and he mentioned the word 'consistency' quite a lot.
"Saka has been nothing short of spectacularly consistent - and consistently spectacular - all season.
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"It's well deserved. I understand a little bit of trepidation with giving a massive contract to a 21-year-old and the way Arsenal have been burned in the past.
"We signed Aubameyang up to a similar deal in 2020 and his form fell off a cliff. He ended up in numerous bits of trouble.
"You can go back to Ozil being given £350,000-a-week to sit in the stands with an umbrella.
"Those two are high-profile cases, but I like to think this is a very different player and very different personality, made of some real stern stuff.
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"He came back from Euros, after missing that penalty, and kicked on. I understand it's a lot of money, but that's football.
"If you didn't pay him that at Arsenal, he'd get it somewhere else."