FIRST LOOK! Khloé Kardashian spotted out with rumoured beau Matt Kemp on the same day she files for divorce from Lamar Odom

PUBLISHED: 01:21 EST, 14 December 2013 | UPDATED: 02:59 EST, 14 December 2013

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Khloé Kardashian just filed for divorce from Lamar Odom on Friday and already she's said to be keeping company with another man.

The 29-year-old reality star is apparently in the blossoming stages of a new relationship with Los Angeles Dodgers baseball star Matt Kemp, also 29.

The pair were spotted on Friday going to the gym, with Matt driving his car and Khloé following in her own SUV.

Pictured: Khloé Kardashian was seen heading to the gym with rumoured new man Matt Kemp in separate cars on Friday - the same day she filed for divorce from Lamar Odom

Right behind you: The reality star was seen driving her white Range Rover a close distance behind Matt in his black Jaguar on Friday

Matt drove at a steady, slow pace, perhaps so Khloé wouldn't be left behind at a stop light.

Khloé kept her eyes shaded behind large-frame sunglasses and just seemed to focus on the road ahead.

A short while later, she pulled into the car park at the gym and walked to the entrance separately while covering her face with her camouflage ruck sack and her beige designer handbag.

Her romance with Rihanna's ex has reportedly been in swing for some time, according to RadarOnline.

Safe pace: The 29-year-old Dodgers baseball player drove at a cautious pace down the tree-lined street in LA

Taking pains: Khloé and Matt seemed to go through a lot of trouble just to go to the gym

Diamond stud: The pro athlete kept the visor down but his diamond earrings still flashed as they caught the light from time to time

On Thursday, Kemp added fodder to the fire when he posted a snapshot of the Kardashian family Christmas tree standing in Kris Jenner's Calabasas home,

'Best time of the year!! It’s gonna b a Cali Christmas this year,' he wrote in the caption.

The responses from his friends and followers came rushing back.

'I'm so happy for you both, wishing you the best,' one pal wrote.

Another chimed in, less favourably: 'Kardahians will do anything for more fame and relevancy. The fact they caught you in their web, I hope you enjoy the rest of your career. '

'Not the kardashians!!!!!!!' echoed a third fan.

Not wasting any time: Khloé is said to be acting like she's already in love, according to reports

Got my eye on you: Matt could probably see his rumoured new lady love following in his rear view mirror

Low profile: Keeping his hands on the wheel, Matt just kept his eyes on the road ahead of him

HollywoodLife.com reports that Khloé already seems smitten with Matt.

'Khloé and Matt are dating, but it’s not serious yet,' a source told the site. 'But Khloé’s already acting like she’s in love... Khloé’s like that; she loves hard, but she moves on quick.'

RadarOnline reported that the pair hung out together in the VIP area at the Drake concert on November 25.

Destination: The reality TV star grabbed her bags out of the vehicle in the gym car park a short time later, using them to shield her face

The great cover up: The brunette lifted her camouflage back pack and beige designer purse high up in front of her as she exited her vehicle

On December 1, Khloé and the Oklahoma-bred athlete both attended the John Legend show at LA's Nokia Theatre.

And then on Monday, they were both at Jay Z's concert in LA, which Radar claims that Lamar also attended separately and he didn't speak to his estranged wife.

Khloé was pictured in an all-black ensemble strolling into the venue in snakeskin high heels and with her head down and clutching her handbag.

New romance: Khloé is said to be getting close to baseball star Matt Kemp; the two were pictured strolling to the Jay Z concert on Monday separately

Low profile: The 29-year-old star kept her head down as she headed to the concert at the Staples Center in LA

Her long brown hair was half pulled back while the rest was allowed to flow around her shoulders.

Matt, meanwhile, was photographed walking to the same venue with a cast on his left leg after breaking his ankle.

In addition to the big smile on his bearded face, he was dressed smartly in black trousers, a white top and beige cardigan, with a driver's cap perched jauntily on his head.

'Best time of the year': Matt fueled romance talk when he posted this picture of the Kardashian family Christmas tree at Kris Jenner's home on his Instagram page

Playing the field: Matt, pictured in 2009, currently plays outfielder for the Los Angeles Dodgers

Former love: Matt dated Rihanna in 2010, when they were pictured at a Clippers basketball game in Los Angeles

The news of the romance comes on the same day - Friday - that Khloé filed for divorce from Lamar in Los Angeles Superior Court, citing irreconcilable differences.

Khloé, who has retained power attorney Laura Wasser, doesn't want spousal support and requests that Lamar not be granted it either.

The couple, who were married on September 27, 2009, have an 'ironclad' prenup that should make division of assets fairly easy.