Is there a cd player that sounds close to an LP?

An honest A/B comparison of analogue to digital is challenging. You need good quality LPs and CDs of the same recording, a phono stage with a separate volumn control (the ear tends to prefer the louder sound, so you have to control for this) and comparable, high-end analogue and digital gear not to mention interconnects. The more neutral, revealing and musical the rest of the system is, the better. In my listening tests, sometimes accompanied by other audiophiles, a Cary 303/200 CDP is compared with a Sota Comet w/Benz-Micro Reference and EAR 834P phono stage w/nos Mazda tubes. While the Cary does sound analogue-like, it is deficient in certain respects. On Sarah Vaughan's "How Long Has This Been Going On?" there is a body and depth to Ms. Vaughn's bass vocals that is simply not imparted on digital. Leonard Bernstein conducting the New York Philharmonic in Copland's Appalacian Spring, Rodeo, etc. sounds marvelous on digital, but the analogue is more dynamic and, again, has a fuller bass. As has been noted on Audiogon and elsewhere, upper end distortion/glare is a problem with many CD transfers. There may also be a loss of naturalness, ambience and air. It has been my experience that there are a few CDs that will not cause ear fatigue. (Trio da Paz's Black Orpheus comes to mind.) This has led me to conclude that most of the sonic loss one typically hears from LP to CD is attributable to something not inherent in the CD format itself.

I'd try a MIT T-2 interconnect on your CDP.
Tone control or not,it may do the trick for you and wont bother the analog set up you have.
Used T-2's are cheap and hold their resale value well in case you dont like it.

Yeah, that's one way to tame cd nasties, throw a wet blanket MIT cable over the signal.

I am not so sure cd's glare. I just had some folks over for an audio party and one of them brought over a Wadia cd player. It went up against my far cheaper Jolida 100. Using the Wadia, there was a noticeable frequency condensing going on. It sounded HiFi. Decay was truncated, and occasionally glare crept in. Because minor detail was lost , clues such as enclosure reverberations and air were lost. Bass retrieval suffered too. With the Jolida, a great cathedral organ sounded just like that, whereas the Wadia reengineered the pipe vibrations into musical notes, devoid of color.

I want the recording of a performance to draw my imagination in. That is what LPs do so well, and that is what my tube outstage cd player does well too.

A few years ago, after I moved back to my mom's house for the summer, I had the loft for my stereo system. The acoustics in the loft was pretty bad. I had to move the system around the den before even the LPs sounded adequate. But, the CD always sounded dry and bright. I then upgraded the Marantz 63SE to CAL Delta/Sigma. Things improved, but the LP still sounded much better.

At the suggestion of DeKay, I removed all the cables and gave all the connections a good polish. I separated all the powercables from signal cables. And, I separated the digital stuff from the amp stuff. Afterwards, even the CD became listenable again.

Dekay is just the man!

No, but I can point you in the direction of a TT whose sound is closer to the positive attributes of a CD: modded Technics 1200 with tonearm fluid damper. Read Zaikesman's review in the accessories section.

It is my goal to keep my CD and vinyl playback close to each other and balanced, performacewise. I had a serious conversation with Dusty Vawter about this subject prior to buying his DAC.

To answer your question directly, you need to attack noise control, damping and isolation in your digital rig. This will assure the smoothest, jitter and artifact free playback possible.