Padme having force powers or being force sensitive is yet to be confirmed. We don't know for sure if she was, however there are some scenes in which she is described to have a certain kind of "intuition." Leia has a similar description for herself, calling it a "hunch." Luke tells her this the Force. Notice these passages in the canon novel, "Bloodline":
Bloodline by Claudia Gray, p49
Leia mentally cataloged that name for later. Maybe her curiosity has only seized upon a random group of people, frequent gamblers who would naturally visit a world like this often, calmer than most. Her suspicion of them was only a hunch, but her hunches often proved to be right. Luke said those strong instincts of hers served as proof of the Force, evidence that it was working through her all the time....
Bloodline by Claudia Gray, p50
Besides, strong as her doubts were, she knew they were based mostly on her---call it intuition....
Padme, on occasion, will mention she has similar "intuitions."
- In The Phantom Menace's script, there is a scene between Padme and Anakin in which Padme finishes Anakin's sentence:
ANAKIN hands a wooden pendant to Padme. She inspects it, then puts it around her neck.
PADME: It's beautiful, but I don't need this to remember you. Many things will change when we Rea h the capital, Annie. My caring for you will always remain.
ANAKIN: I care for you too. Only I...miss..
ANAKIN is disturbed by something. Tears are in his eyes.
PADME: ....You miss your mother
- Several lines were cut out of the beginning of the script in "Attack of the Clones," but Padme has a "gut feeling" that Count Dooku was behind the attempts on her life.
YODA: With you, the Force is strong...young Senator. To see you alive, brings warm feelings to my heart.
PADME: Thank you, Master Yoda. Do you have any idea who was behind the attack?
MACE WINDU: Our intelligence points to disgruntled spice miners, on the moons of Naboo.
PADME: I don't mean to disagree but I think that Count Dooku was behind it.
There is a stir of surprise.
MAC WINDU: You know, M'Lady, Count Dooku was once a Jedi. He wouldn't assassinate anyone, it is not in his character.
KI-ADI-MUNDI: He is a political idealist, not a murderer.
YODA: In dark times nothing is what it appears to be, but the fact remains, Senator, in grave danger you are...
PALPATINE: Count Dooku has always avoided any kind of conflict. It appears he has no desire to start a war. Why would he kill you? To what end?
PADME: I don't know, but everything in my being tells me he was behind it...
- In a deleted scene in "Attack of the Clones", Padme notices Anakin is having a nightmare, so she wakes him up and somehow knows he was dreaming about his mother:
PADME (changing the subject): You were dreaming about your mother earlier, weren't you?
ANAKIN: Yes...I left Tatooine so long ago, my memory of her is fading....
- In the novelization and offical script (which was cut out) of "Revenge of the Sith," Padme has a feeling she is carrying a son, though Anakin thinks it is a daughter:
ANAKIN: How are you feeling?.
PADME: He keeps kicking.
ANAKIN: He?! Why do you think it's a boy?
PADME (laughs): My motherly intuition
She puts his hand on her belly
ANAKIN: Whoa! With a kick that strong, it's got to be a girl.
Anakin and Padme seem to sense the other from across the city in "Revenge of the Sith."
There was again, several lines that were cut out of the offical script of "Revenge of the Sith" that again seem to tell us that Padme has some force - sensitivity:
PADME: What is it?
PADME: Don't do this...don't shut me out. Let me help you.
ANAKIN: You can't help me...I'm trying to help you.
They look in each other's eyes.
ANAKIN (continuing): I sense...there are things you are not telling me.
Padme is startled at this.
PADME: I sense there are things you are not telling me.
Padme smiles. ANAKIN is a little embarrassed.
- Padme knows there is still good in Anakin at the time of her death, just as her son, Luke will later believe.
The Art of Star Wars Episode III: Revenge of the Sith
In the book, The Art of Star Wars Episode III: Revenge of the Sith, it was revealed that Padme had more midi-chlorians than Anakin during her pregnancy with Luke and Leia. Several pieces of concept art show this as well. According to Iain McCaig regarding the story concept and artwork:
"There are radical transformations in a woman during normal pregnancy: how much more intense would it be if her twin embryos were also filled with Force-laden midi-chlorians? Three unused story concepts: Padmé’s waking visions of Mustafar: Padmé’s sleepwalking manifestations of the Force: Padmé’s agony as her body reacts against the midi-chlorians."
"Padme has the Force flowing through her. She has more midi-chlorians than any person ever had because she’s pregnant with the Skywalker twins, so I was wondering if we see her suffer."
George Lucas was also enthusastic about this idea before it was scrapped from the film
"...Padmé had more midi-chlorians than Anakin during her pregnancy with Luke and Leia, having the Force flowing through her as she doubled up in pain seeking Yoda's help as seen in an sketch. The concept wasn't used in the final cut despite George Lucas' enthusiasm over the idea."
The concept art shows Padme having visions of Anakin being rescued by droids on Mustafar, "the sleepwalking manifestations of the Force", levitating rocks, and Padme doubled over in agony while Yoda is unable to help her.
There are still little hints here and there left in canon that suggest Padme is force sensitive (more so when she is pregnant with the twins). It seems that a lot was cut out of the films (probably due to screen time) but since there is no confirmation on the subject, it left to debate and opinion.