According to the Daily Mail's Dominic King (h/t Uche Amako of the Daily Express), the Reds were "happy to let" Sanchez join Tottenham Hotspur from Ajax, while Napoli star Koulibaly was deemed too expensive.
Liverpool were closely linked with Van Dijk throughout the summer, and despite the defender handing in a transfer request in August, per the Guardian's Mark Dobson, his firmest admirers failed to make their move.
Sanchez eventually moved to Tottenham in a deal worth a reported club-record sum of £42 million, and football writer Jordan Clarke enthused what a positive arrival the Colombia international has been in north London:
While the fee paid for Sanchez's signature may seem extravagant for a 21-year-old, the central defender holds great promise and is already acclimating well to the physical demands on the Premier League.
Not only that, but Liverpool manager Jurgen Klopp has seen the youngster excel under Spurs boss Mauricio Pochettino, and he demonstrated a more nuanced side to his game in Saturday's 4-0 thrashing of Huddersfield Town, per Squawka:
Koulibaly would have required a much greater investment from Liverpool, and Napoli president Aurelio De Laurentiis told Sky Sports he rejected Chelsea's £58 million offer for the player two days before the transfer window closed.
That gives some sense as to the kind of outlay the Senegal international would require if he were to leave the Naples outfit. Napoli have since embarked upon a perfect start to the season and lead Serie A, with Koulibaly's agent, Bruno Satin, recently telling Radio Crc (h/t TalkSport) his client is content at the club:
"How much is Koulibaly worth? I don't know, in the most recent summer some of the rules of the transfer market changed.
"We renewed his contract a year ago and it's valid until 2021. He's calm and happy, he's playing for Napoli in a great team which is top of the league.
"It's been a long time since Napoli have played such fun and spectacular football, they score so many goals and you can see that the players love playing in this team."
Defence was known to be an area in need of reinforcements at Anfield, and after Andrew Robertson was signed from Hull City to provide cover at left-back, recruiting a centre-half should have been a priority for Klopp.
But their failure to add any new names after missing out on Van Dijk has left Joel Matip and Dejan Lovren as the central pairing of choice. Matip is viewed as the more capable of the two, but Liverpool writer Leanne Prescott recently said even the Cameroonian has his flaws:
Liverpool's defensive flaws were exposed once again in Sunday's 1-1 draw at Newcastle United, where Joselu evaded his markers too easily to equalise. His goal was the 12th the Reds have conceded in the Premier League this term.
While Sanchez and Koulibaly go from strength to strength at their respective clubs, Liverpool fans have been left to settle for more of the same and await the January window as their next chance to freshen up the back line.