The 35-year-old, who joined Spurs in 2012, was free to talk to foreign clubs from the start of this month but has committed his future to Tottenham in a deal that runs until 2024.
AC Milan full-back Hernandez netted his first France goal in the 90th minute of his second international appearance after France had come back from 2-0 down at half-time at the Allianz Stadium in Turin through strikes from Karim Benzema and Kylian Mbappe.
Giroud, who turns 35 next month, misses out despite sitting just five goals away from equalling Thierry Henry's all-time record tally of 51 for Les Bleus.
Euro 2020: 'What a game. What a tournament' — Twitter reacts to Switzerland and Spain's dramatic wins
Switzerland eliminated world champions France with a penalty shootout win while Spain knocked Croatia out with help of extra time goals.
Euro 2020: 'It's painful', says France captain Hugo Lloris after shock Round of 16 defeat against Switzerland
Didier Deschamps' team fell behind in Bucharest on Monday but Karim Benzema scored twice before Paul Pogba added a third to leave France seemingly on course for the quarter-finals.
Euro 2020: Kylian Mbappe misses decisive penalty as Switzerland stun world champions France in Round of 16
Mario Gavranovic equalised in the 90th minute as Switzerland came from two goals down before Yann Sommer saved Mbappe's penalty to book Switzerland a quarter-final clash against Spain.
Euro 2020: France captain Hugo Lloris warns world champions against over confidence ahead of Hungary clash
Tottenham goalkeeper Lloris says Les Bleus need to prepare themselves for a challenge different from the one against Germany on Saturday in front of a big crowd at the Puskas Arena in Budapest.
Even the generally pragmatic Deschamps struggled to contain his enthusiasm when asked recently about a forward line featuring Benzema alongside Kylian Mbappe and Antoine Griezmann.
Aymeric Laporte's header eight minutes from time finally ended Tottenham's resistance as their 13-year wait to win a trophy goes on.
FIFA World Cup qualifiers: Ousmane Dembele scores, Kylian Mbappe misses penalty as France beat Kazakhstan
It was a welcome result for reigning world champions France after they kicked off their campaign in qualifying Group D by drawing 1-1 at home to Ukraine last Wednesday.
FIFA World Cup 2022 Qualifiers: Familiar look to Didier Deschamps' France for start of title defence
Only 14 members of Didier Deschamps' 26-man squad for the start of qualifying went to the last World Cup
Captain Hugo Lloris had strongly criticised the underlying culture at the club in a stinging interview following Thursday's loss in Croatia, pointing the finger at fringe players who were not fully committed.
Mourinho said the 34-year-old remained one of the best goalkeepers in the world, comparing his errors with those of Liverpool's Alisson Becker.
For the first time in Mourinho's managerial career, he has lost back-to-back home league games.
The Spurs striker was sorely missed during Sunday's dismal 1-0 defeat at lowly Brighton after damaging both ankles in last week's 3-1 loss to champions Liverpool.
Jose Mourinho's side climbed above previous leaders Liverpool on goal difference after a sterile London derby at Stamford Bridge.
World Cup winner Pogba has been replaced by teenager Eduardo Camavinga, who was called up for the first time alongside Lyon midfielder Houssem Aouar and RB Leipzig defender Dayot Upamecano.
Hart will compete with Paulo Gazzaniga to sit on the bench — and get the nod for either League Cup or Europa League ties — as understudy to France's World Cup winning goalkeeper Hugo Lloris.
Joe Hart signed a two-year contract with Spurs with the club reportedly agreeing to pay him £50,000 ($66,000) a week, the same as his wages in an unsuccessful spell with Burnley.
Premier League: 'Beautiful' Hugo Lloris-Son Heung-min row a sign Tottenham are growing up, says manager Jose Mourinho
Goalkeeper Lloris and forward Son had to be held apart by team mates shortly after Everton’s Richarlison had gone close to an equaliser, with Lloris clearly angry at Son for not tracking back