Like so many young players at Stamford Bridge, the 21-year-old has found the pathway to the first-team blocked and has been farmed out on loan.
Bamford spent successful spells at MK Dons and Derby last season, before linking up with Middlesbrough this term - a move which has seen the striker's game go up a notch.
Playing under Aitor Karanka, Jose Mourinho's assistant at Real Madrid, the striker has netted 17 goals in Boro's promotion push, with those performances seeing him crowned Championship Player of the Year on Sunday evening.
It is an accolade which will no doubt catch the eye of those at Chelsea, where Bamford remains determined to make his mark.
"At the start of the season, I had a discussion with Chelsea and they decided that out of the clubs that were interested, Middlesbrough would be the best fit," the England Under-21s international said.
"Not only for me as a player but it would make me ready for Chelsea next year or the year after.
"At the minute, there's no reason why I wouldn't want to be at Chelsea
That's my dream, to pull on that blue shirt, so hopefully it'll come true.
"It will be a tough task to break into the first team
Most strikers playing in the Premier League regularly are aged around 23 or 24.
"At a club like Chelsea they have the money to buy world class strikers like Diego Costa and there's also that pressure to win trophies, I just have to stick with it, be patient and hopefully my rewards will come."
Bamford clearly sees his long-term future at Chelsea but, while negotiations to extend his contract are ongoing , accepts another loan spell may be beneficial next season
A return to Boro would make sense, especially as it looks like the Riverside Stadium may soon be hosting Premier League football.
Friday's vital 1-0 win at Norwich means only goal difference separates Boro from an automatic promotion spot with two matches remaining.
"You can see the work we put in week-in week-out on the training ground and the work everyone puts in," Bamford said.
"We do deserve to go up but I think a lot of teams think the same thing
It's going to be tight, it's going to be an exciting finish to be sure."
It is a run-in Bamford hopes to be involved in, having sustained an ankle injury in the win at Carrow Road.
The striker somehow battled on but is a doubt for the remaining matches against Fulham and Brighton.
"I've got a scan (scheduled) and it's a bit sore but I'll try and play on," Bamford said
"It was massive win for us at Norwich on Friday
It wasn't our best game of the season, but we have the best defensive record and you saw why."
Source : PA
Source: PA