The Dutchman, 53, guided Southampton to their highest Premier League points total of 63 thanks to the 4-1 victory over Crystal Palace on Sunday.
Koeman's contract at St Mary's runs until the summer of 2017.
"It's one thing speaking about ambition and the second is to show it," Koeman told BBC Radio Solent.
"It's all about trying to improve and to continue. And of course now the expectations will be higher than last season.
"Last season was already a special one and to do even better than last year is incredible."
The former Barcelona defender has been linked with the vacant managerial role at Everton but has said he expects to see out his contract at St Mary's,
Southampton are fifth in the Premier League, although Manchester United will climb above them if they avoid defeat in their rearranged match against Bournemouth on Tuesday.
Saints are guaranteed Europa League football for the second season in a row.
Southampton manager Ronald Koeman has urged the club's board to prove its ambition as he prepares for talks about a new contract this week.