Saturday's 3-0 defeat by Newcastle United had left them a place and a point above the relegation zone and sealed the Argentine's fate.
Klopp, who has yet to win a trophy as Liverpool manager since being handed the job in 2015, saw his men go down 2-1 at Manchester United on Saturday to make finishing second in the Premier League, above the Red Devils, a tough ask. He [Rashford] played for Manchester United many, many, many times.
The Portuguese added: "I try to give the best to the kids, education".
Southampton have lost a large number of players to their domestic rivals in the past few years and club chairman Ralph Krueger said earlier in the season that they were keen to end the perception that they were a selling club.
Jürgen Klopp's side responded in the second period, pulling one back by forcing Eric Bailly to put through his own net just past the hour mark.
"If you leave the technical area by a couple of metres - or in this case 20 metres - but to speak to your own player and not to complain about the referee's decision, then they made the right decision".
I feel like both teams are more bothered about not losing than winning, but I fancy Liverpool to break through United's backline, open the scoring and jolt United into some attacking play.
Inflation falls to 4.4 per cent, factory growth rises
The RBI targets inflation over the medium term at 4 percent with an upper limit of 6 percent and a lower threshold of 2 percent. The manufacturing sector saw growth quickening marginally to 8.7% in January from 8.4% in the previous month.
Rashford was the local boy who had just scored a quite superb...
"In the situations around the two goals, it was not like it should be - you can not leave a player alone", assessed Klopp.
"Liverpool pushed us into defensive situation but we kept control".
Liverpool found themselves two goals down at half-time after two goal-kicks that were won superbly by Romelu Lukaku were twice punished by Rashford.
In the North East, Newcastle United secured a 3-0 victory over Southampton to move five points clear of the bottom three, piling the pressure on Saints boss Mauricio Pellegrino.
"Obviously to concede two nearly identical goals was frustrating and it was hard to break them down today because they sat back waiting for the counter-attack".
West Bromwich Albion's woes continued as they slipped to a 4-1 home defeat by Leicester City.