Goalkeeper Davies agreed to join the Potters more than a month ago while centre-back Lindsay has been brought in for £2million which could rise to £2.5million.
They were the two players in the Barnsley squad who had played the most games for the Reds while they each appeared in more than 40 League One matches during the promotion campaign.
While Stoke are in the same division as the Reds, both players are understood to have significantly increased their salaries by moving to the Midlands club.
Wales international Davies, who captained the Reds last season, played 209 games for the club he joined in 2014 and helped them to two promotions. He was the oldest player at Oakwell aged 26.
Scottish defender Lindsay, 23, played 90 games after joining from Scottish club Partick Thistle in 2017. He has been a regular in both the relegation season from the Championship then the following promotion.
Barnsley have already brought in goalkeeper Brad Collins and Danish centre-back Mads Andersen, while they are likely to recruit further players in both of those positions.