The former Hammers boss was a guest on talkSPORT when he recalled receiving word of the chance to sign the burly striker, famous for scoring Portugal's brilliant extra-time winner in the European Championship final last summer.
Allardyce wanted back-up for the injury-prone Andy Carroll at the time and felt Eder could 'do a job for us'.
The opportunity came about to take the player on loan and Big Sam wasted little time jetting out to Portugal with his chief scout at the time to take a look at Eder in action and seal the deal.
Everything seemed to be going to plan and Allardyce was sufficiently impressed by the player's performance.
Eder was keen too and had even packed ready for East London not long after the final whistle.
But when Allardyce met the player and club officials in a hotel things took an unexpected twist.
"We arranged to meet up in a hotel after the game ... with the president of the club and the secretary of the club and the chief executive of the club, the player came along, the English agent, myself and the chief scout sorted out the loan deal.
"The lad came with his bag to travel with us the day after and I got a phone call from (former QPR player) Mick Walsh, who lives in Portugal, and he said 'Have you got Eder there?'
"I said 'Yea' and he said 'Well are you dealing with his agent?' and I said 'Yea he's here' and he said 'No that's not the right agent Sam'. I said 'Walshy it all seems like it's done, it all seems ok'.
"So we sat around the table and all of a sudden we see Eder look up and his eyes started staring so we turn around and there's three big guys at the door just waving to him, do you know what I mean?
"So Eder goes to meet these guys and that was the last we saw of him, he'd gone.
"The owner of the club was going mad, we stayed up until about 3am, we called the police, we called everybody and that was it, we got up the next day with the agent, traveled all the way back to England and we never saw hide nor hair of him again until he turned up at Swansea. I was just relieved he was ok."
So there you have it West Ham fans, the man who netted the winner for Portugal at Euro 2016 could have been wearing the claret and blue.
An ill-fated spell at Swansea did follow for the striker, who now plies his trade in Russia.
Eder isn't the only Portuguese star the Hammers have missed out on signing, though.
Slaven Bilic reportedly turned down the chance to sign Renato Sanches this summer while moves for Barcelona's Andre Gomes and William Carvalho failed.